l I I I I I I I IlIlIlllIII:

llllllllllllllllllllllllllllllllllllllllllllllllllllllll

31293 02058 9861

This is to certify that the

dissertation entitled

New Elements of Analysis on the Levi-Civita Field
presented by
Khodr Mahmoud Shamseddine
has been accepted towards fulfillment

of the requirements for

dual PhD degree in Physics and Mathematics

 

 

Heel/(x.— 5%

Major professor

Date December 10, 1999

 

MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771

 

Lia-mm
Michigan State
University

 

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINE return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

moo chlRC/DathHpGS-afl

 

New Elements of Analysis on the Levi-Civita Field

By

Khodr Mahmoud Shamseddine

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics
and
Department of Physics and Astronomy

1999

O
\' ‘
.‘ ‘ ‘_ t.‘ O
Pi ntijhfnt‘

‘“ ‘lel'i'mm
~A ~-
\ ‘é, a

ABSTRACT

New Elements of Analysis on the Levi—Civita Field

By

Khodr Mahmoud Shamseddine

New elements of analysis on the Levi-Civita field R are presented. First we prove
general results about skeleton groups and field automorphisms that will enhance the
understanding of the structure of the field. We show that while the identity map is
the only field automorphism on R, there can be nontrivial automorphisms on non-
Archimedean field extensions of R like R. We also show that every automorphism on
R is order preserving and that if P is such an automorphism and r a real number then

P(r) is approximately equal to r; moreover, if q is a rational number, then P(q) = q.

After reviewing the algebraic, order, and topological structures of the field ’R. [3, 5,
7], we review two types of convergence and prove new results about the convergence of
the sums and products of sequences and infinite series. A weak convergence criterion
[5] for power series is then enhanced and proved, and we show that power series
can be reexpanded around any point of their domain of convergence. Knowledge
of weak convergence of power series allows the extension to the new field and the
study of all transcendental functions. This also will allow the extension of all the real
functions that can be represented on a computer and is thus of great importance for

the implementation of the ’R, calculus on computers [38, 39, 40, 42].

We review two different definitions of continuity and differentiability [5, 10]. We

show that thase smoothness criteria are preserved under addition, multiplication and

‘ ' I - w;
"canon of rum t.
“.00 -~

A

- - O . ‘ .‘.
fl. o-"r, “Que I I v ‘
‘1 wthU Adt’hllt} ‘
"We“ mmon l
L' ‘.. -u k -h

. , 1 r
a" ‘ inch” ows :t
.v‘ t‘p‘; quai‘rlgpr C(IIIIF'
'6 --- ‘- 5 - “-3" _
.. ' ‘ .
*‘ztzcm tor ml?

‘v AoCW LmCt:(:'n

_"‘~¢-~ A

Bi‘t’fi on 0m k.

:5 5273:1115 Whlffl 1
$2225- the nor:z..
52:20:15 :41. l
:5 In particular
3?: and the mi-az.
ii itegable: and

‘.’."‘
\

“""bv-

7'1
'c ..‘- _
‘“‘ mime 01

;’ ‘v, =
c n v
. N1

I "3‘, i ‘
|~‘l‘ nil-TL] ‘

n. ‘ ~

nl :‘ a,

.‘i~‘t‘pr‘:

"‘li‘able P9

composition of functions. We show with several examples that topological continuity
and differentiability are not sufficient to assure that a function be bounded or satisfy
any of the common theorems of real calculus on a closed interval of R. We derive a
result which allows for an easy check of the differentiability of functions. Then, based
on the stronger concept of differentiability, we present a detailed study of a large class
of functions for which we generalize the intermediate value theorem in [5] and prove

an inverse function theorem.

Based on our knowledge of convergence of power series, we study a large class
of fimctions which are given locally by power series with R coefficients and which
generalize the normal functions discussed in [5]. We show that the so-called expand—
able functions [41, 43] form an algebra and have all the nice properties of real power
series. In particular, they satisfy the intermediate value theorem, the maximum the-
orem and the mean value theorem. Moreover, they are infinitely often differentiable
and integrable; and the derivative functions of all orders are themeselves expandable

functions.

The existence of infinitely small numbers in the non-Archimedean field 7?. allows
the use of the old numerical algorithm for computing derivatives of real functions,
but now with an error that in a rigorous way can be shown to become infinitely small
(and hence irrelevant). Using calculus on R, we formulate a necessary and sufficient
condition for the derivatives of real functions representable on a computer to exist
at any given real point, and we show how to accurately compute the derivatives up
to very high orders if they exist, even when the coding exhibits branch points or

nondifferentiable pieces [38, 39, 40, 42].

 

COW. l ‘.
t-T‘Ern

Midi \t ‘

~ ‘d‘l

1 -
$39

 

Copyright by
Khodr Mahmoud Shamseddine
1999

 

To my parents Latifa and Mahmoud

.zxgnmnthe}t

l I I

. , I
51"?“ r9: P! '1'
.. .i..p (1.10 3 .,
l"

a 1': [Zippl'rrtuzxzt '

-.- spec me pm.

Li -E‘J U‘Ddf‘l‘ U-j “N
e O ‘

9:19 y i ' ‘
.. ,_ ‘ I, - u” - ' '
u ‘ t .-tl-cJ\‘AI-J.u‘l.

Professional; Ft

rr'

.
‘o.
,.i'.-
o,.‘i
‘ \

. o '
l 3‘ ‘3"! N V 0
‘ldhl; AI‘i‘t

w; »

‘ ~§fif1fiy to Wii

‘ .
i

:1; :1
magi). and P
‘ 1991.“,
1 .IL'Q‘ . P
:‘ glq‘
g
I,
‘ 'JE‘ n“'.‘l ‘
‘ ’1‘ Pal]
\-\:~S aid :( L
1Jr (J‘JIY
in”.
‘\‘1 ‘v
“H?"

ACKNOWLEDGMENTS

Throughout the years of my graduate study at Michigan State University, I have
received help and support from many people and organizations; and I would like to
use this opportunity to express my thanks and appreciation to all of them. Some
have helped me professionally, and others have touched my life at the deepest levels
and their impact goes far beyond their support for me in finishing my dissertation. In
order not to confuse the extent of my gratitude to the various people, I will distinguish

between professional and private acknowledgments.

Professional: For introducing me to the exciting field of Non-Archimedean Anal-
ysis; for fostering my love of Mathematics, its beauty, clarity and rigor; for supporting
me financially to work on my research and to go to conferences both in the United
States and abroad; for providing me with many useful discussions, ideas, suggestions,
and corrections; and for helping me at the personal level on many occasions, I would
like to thank my advisor Prof. Martin Berz. I would like also to thank Prof. Christian
Bischof, Prof. Jerzy Borysowicz, Prof. James Linnemann, Prof. Jerry Nolen, Prof.
Joel Shapiro, and Prof. Vera Zeidan who kindly accepted to serve on my committee.
I am especially grateful to Prof. Zeidan for helping me, together with Prof. Berz,
With the dual PhD arrangement between the Physics and the Mathematics depart-
ments and for being a supportive friend. I am also thankful to Prof. Linnemann

for motivating me to add Sections 1.3, 1.4 and 1.5 which prepare the reader for the

vi

'6“ ""2-«F W]

 

;-r"’atit:t1£0l In.V ““1

ii: an of finamri
:sgiiial {harks g0
31'. its. Christian
‘La Rabat. and l

. .
- as Batman.

carat mt man

the «titling in
Eris iith whom I i
rii file to thank l
in. Balaniin. 5
32a: lindemann
3: dictating div-
r to Jens H

A...

O“‘u? 3:?

.. ‘v-_ I “7" .
TL 'M 5 m‘n

it»;

Live-Hg: my St a\v I
‘3. :Va 2
m... .‘ .931 \mzou;\ I)

.sjsiafiy thank l

q¢iJ3Hartu g

. ~~~~~~

applications of my work in Physics, discussed in details in Chapter 7.

For years of financial and moral support, I am indebted to the Hariri Foundation;
my special thanks go to H.E. PM Rafic Hariri, Dr. David J. Thompson, Mr. Rafic
Bizri, Mrs. Christiane Gangi Buck, Mr. Joe Murnan, Mr. Mustafa Zaatari, Dr.
Wadih Kanbar, and Dr. Abdul-Rahman Sidani. I am also indebted to the Alfred
P. Sloan Foundation, and the United States Department of Energy for financially

supporting my research for the last five years.

While working in the group of Prof. Berz, I had the opportunity to meet talented
people with whom I had nice friendships and very enjoyable scientific discussions. I
would like to thank Dr. Kyoko Makino, Dr. Georg Hoffstatter, Dr. Weishi Wan, Dr.
Vladimir Balandin, Silke Rolles, Jens Hoefkens, Bela Erdelyi, Jens von Bergmann,
Michael Lindemann and Lars Diening. I am especially grateful to Kyoko for the
many educating discussions I had with her over the years about science and human
behavior, to Jens Hoeflrens for his continuous help with computer problems, and to

Jens von Bergmann for proofreading much of this work and giving useful suggestions.

During my stay at Michigan State University, I received support and encourage-
ment from various people in the three departments I was affiliated with. I would like
to especially thank Prof. Peter Lappan, Prof. Clifford Weil, Barbara Miller and Ster-
ling Tryon-Hartwig from the Mathematics department; Shari Conroy, Kay Barber,
Jean McIntyre and Chris Townsend from the National Superconducting Cyclotron
Laboratory; and Prof. Julius Kovacs, Stephanie Holland and Debbie Simmons from

the Physics and Astronomy department.

Finally, I would like to register here my admiration and appreciation to all the
great minds and souls who throughout history have enriched humanity with Knowl-

edge, Beauty, Peace and Love and who have thus made this world a better place and

vii

_..: Trim more an
921591: such great

“age; mu work )‘0'

21:}:“6 He expat
3:535 rest a kinri
:52: for '15. restrit;
2435110 us. O‘lr

. J . ~
M. Di CL‘KLllfipxtt‘

‘\ Albert E.)

?
ELIE"l"'np -»
Mano. )0

11:2 a
. «15 me e I
.m .
tilip‘rdt ,
‘63:, -
”Cu"?
““3 Da

made mm more accessible to us. I would like to recall four inspiring quotations
from four such great minds of the twentieth century: a poet and artist, a scientist,

and two peace advocates.

“When you work you are a flute through whose heart the whispering of the hours

turns to music.” Kahlil Gibran.

“A human being is a part of the whole called by us universe, a part limited in time
and space. He experiences himself, his thoughts and feelings as something separated
from the rest, a kind of optical delusion of his consciousness. This delusion is a kind of
prison for us, restricting us to our personal desires and to affection for a few persons
nearest to us. Our task must be to free ourselves from this prison by widening our
circle of compassion to enhance all living creatures and the whole of nature in its

beauty.” Albert Einstein.
“Be the change you want to see in the world.” Mahatma Gandhi.

“I ask you one thing: do not tire of giving, but do not give your leftovers. Give until

it hurts, until you feel the pain.” Mother Teresa.

Private: Special reverence and appreciation go to my parents. Besides raising me
and giving me all the love and support I needed since I was born, they have always
been an inspiration for me and my brothers through their hard work, dedication,
mH—deniance, patience, optimism and faith in Life. They taught me that a good
education and strong moral values are the longest lasting wealth one can acquire in
one life time; they have always supported me and they continue to support me in my
goal to become a dedicated scientist. I am so grateful that my father has prevailed in
a recent fight against a serious sickness that kept him hospitalized for over two and
a half months and at times threatened his life. I am proud of him and of my whole

family who stayed strong by his side all the time.

viii

Fi'l' bainfl With 2

“‘0
,.'.'-vr *n and JD}. t
f. ”3.65111 to Pd?

thug-“I
. .

Efiwmfi&

“9

u .r 'V
“ ‘ ‘ ' o ' .'
W33 mt Can. it?
attends. T
iris for me. I Witt»:
w ' ‘
7T7? mt'bmtners. .
as. lean. Sillldtl
r: Eanien; my l( m
l .L “
4:..2. .llmmujd :4!
”iv. . a ‘ l I ' I'
‘53 Spt’fla .lll‘ilf

airs it element arj

w;
it

(till): I would 1

ignite: Sanira wl

V
5}"

31.. Brian. Sue. .\l

11';
I I

«i. Maria and Rh

t, .

rizeia's. Veronica.

ll“...
«a and Rain. v

”a IOI two tears.

For being with me for two and a half years, giving me love and support and
adding fun and joy to my days and nights; and for enriching my life in many ways,
I am grateful to Patricia Tsune Ballard. I am also thankful to Patricia’s parents,
Yaeko and James Ballard, who welcomed me into their family and were always nice

to me.

Since my childhood, I was fortunate to have around me loving and supportive
family and friends. Their love, support and encouragement have always been a driving
force for me. I would like to thank my aunt Zeina who has always been like a mother
to me; my brothers: Mohammad, Ali, Youssef, Hassane and Saeed; my sisters-in-law:
Fatima, Iman, Suhad, Sana and Nadia; my cousins: Nazek, Abdul-Kareem, Jamal
and Fawzieh; my lovely nieces and nephews: Dalia, Mahmoud, Rana, La Rose, La
Reine, Mahmoud Saeed, Sally, Mohammad and Tareq; my neighbors and relatives;
my very special friends: Mustafa, Nasseem, Bassem, Issam and Mahmoud; and my

teachers in elementary school, high school and at the American University of Beirut.

Finally, I would like to thank my MSU friends who made my stay here more
enjoyable: Samira who has been a close and supportive friend to me for over ten
years, Brian, Sue, Munjed, Joe and Marie, Mireille and Jorge, Rabi, Cathy and
Walid, Maria and Riad. I also would like to thank Zakir, Ahmed, Chen, the two
Gabriella’s, Veronica, Melvin, Jamal, Lulu, Leps, Eric, Oogie, Jose, Gaston, Heather,
Jill, Mika and Radu, with whom I have enjoyed playing volleyball twice or three times

a week for two years.

Contents

llntroduction

1.1 Motivation.
I2 Outline. . .
l3 ht‘litiw lie:
14 Applicanons
1.5 Implements:

1.9 Notations

1‘ N
l :leleton Group
ii '

. bteieton Gin-

‘t'l :.1
Feltiitttoni

—o

l The Non-Archit
ll AlgebraicStt
3.3 Di‘ierential .~‘
313 Order Strict

34 leptztlngit'al E

l R
~9quenees and c
n t. k
Siluflg Com-‘-
. n»-
f-di Center
' Power Series

' l
‘

"‘ Time

,5

r“
w

endent

5 r
Lakulus OH R

f l _
TWA-921ml (

I" n‘ilmgg DJ:

, iu‘jI
~ u Pillage

Contents

Introduction

1.1 Motivation .................................
1.2 Outline ...................................
1.3 Intuitive Remarks .............................
1.4 Applications in Physics ..........................
1.5 Implementation ..............................
1.6 Notations .................................

Skeleton Groups and Field Automorphisms
2.1 Skeleton Groups ..............................
2.2 Field Automorphisms ...........................

The Non-Archimedean Field 7?.

3.1 Algebraic Structure ............................
3.2 Differential Algebraic Structure .....................
3.3 Order Structure ..............................
3.4 Topological Structure ...........................

Sequences and Series

4.1 Strong Convergence ............................
4.2 Weak Convergence ............................
4.3 Power Series ................................
4.4 Transcendental Functions ........................

Calculus on R

5.1 Topological Continuity and Topological Differentiability ........
5.2 Continuity and Differentiability .....................
5-3 n-times Difierentiability .........................
5-4 Intermediate Value Theorem and Inverse Function Theorem .....

X

CDmiht—‘H

13
18

19
19
23

32
32
43
45
49

57
57
69
82
89

98
99

115
128

Lipandable Ru
6.} Definition a'.
II.“ Calculus. on'

6.2.1 Inter:
6.3.2 llttti.
6.2.3 Hull"

'33 Integration .

.' Computer Func
Tl Introduction
3 Computer F‘.
5.3 Comping-inn:
7.4 Examples

'3 03319.1(” Fl

6 Expandable Functions 167

6.1 Definition and Algebraic Properties ................... 167
6.2 Calculus on the Expandable Functions ................. 174
6.2.1 Intermediate Value Theorem ................... 175
6.2.2 Maximum Theorem and Mimimum Theorem .......... 185
6.2.3 Rolle’s Theorem and the Mean Value Theorem ......... 188

6.3 Integration ................................. 190
7 Computer Emotions 194
7.1 Introduction ................................ 194
7.2 Computer Functions of One Variable .................. 197
7.3 Computation of Derivatives ....................... 205
7.4 Examples ................................. 211
7.5 Computer Emotions of Many Variables ................. 214

Chapter

lntroduc

1.1 Motive

'-

..t .66. numbers

v—n ' I

.1: 57,):71‘33. pm]

I'W‘lnsm .
.-.-..i.ore. the":

a

2 re enraged a

54.,

_::t§ “3'1; ‘1‘ ‘ ‘
I \.. ,L‘Ln .l‘

.g'r'r». . ‘

:‘-‘r’-m lll'dl 31

Hr‘
, L
‘s

‘:"t‘ fur. :-
u..’ll.l'el;(_vn 0

Chapter 1

Introduction

1.1 Motivation

The real numbers owe their fundamental role in Mathematics and the sciences to
certain special properties. To begin, like all fields, they allow arithmetic calculation.
Furthermore, they allow measurement; any result of even the finest measurement
can be expressed as a real number. Additionally, they allow expression of geometric
concepts, which (for example because of Pythagoras) requires the existence of roots,
a property that at the same time is beneficial for algebra. Furthermore, they allow
the introduction of certain transcendental functions such as exp, which are important
in the sciences and arise from the concept of power series. In addition, they allow the
formulation of an analysis involving differentiation and integration, a requirement for

the expression of even simple laws of nature.

While the first two properties are readily satisfied by the rational numbers, the
geometric requirements demand using at least the set of algebraic numbers. Transcen-
dental functions, being the result of limiting processes, require Cauchy completeness,
and it is easily shown that the field of real numbers is the smallest totally ordered
field having this property. Because it is at such a basic level of our scientific language,

hardly any thought is spent on the fundamental question of whether there may be

1

"sin number

,1; question is
tie the field of r91
gag R" have cert a

. , 1r ‘
3::ertatzea; tornru.
35????ng Intuitive
22:: of improper ft;
7156 within the In
argrcu fashion. it
fine concept of
SEEK scientists sat
. that is sit.
-:a tenor be forr
”v OI computer:

T“I:_L.4"“‘. . ‘
a mud or
‘ A

{3pr ' l’ -
. "lmulplefl All} ‘1

lie- problerns m‘
to the real or
rivers; that is if th
:‘gin complete fit."
T13: males the real 1

Na fanny It .
'3 ‘ bQIllllr
., 3 h .
t’QQIS 0n III“
2‘».
"‘«l j».

other useful number systems having the required properties.

This question is perhaps even more intriguing in light of the observation that,
while the field of real numbers R and its algebraic completion 0 as well as the vector
space R" have certainly proven extremely successful for the expression and rigorous
mathematical formulation of many physical concepts, they have two shortcomings in
interpreting intuitive scientific concepts. First, they do not permit a direct represen-
tation of improper functions such as those used for the description of point charges; of
course, within the framework of distributions, these concepts can be accounted for in
a rigorous fashion, but at the expense of the intuitive interpretation. Second, another
intuitive concept of the fathers of analysis, and for that matter quite a number of
modern scientists sacrificing rigor for intuition, the idea of derivatives as difierential
quotients, that is slopes of secants with infinitely small abscissa and ordinate differ-
ences cannot be formulated rigorously within the real numbers. Especially for the
purpose of computational differentiation, the concept of “derivatives are differential
quotients” would of course be a remedy to many problems, since it would replace
any attempted limiting process involving the unavoidable cancellation of digits by

computer-friendly algebra in a new number system.

The problems mentioned in the preceding paragraphs might be solved if, in ad-
dition to the real numbers, there were also “infinitely small” and “infinitely large”
numbers; that is if the number system were non-Archimedean. Since any Archimedean
Cauchy complete field is isomorphic to R, it is indeed the absence of such numbers
that makes the real numbers unique. However, since the “fine structure” of the con-
tinuum is not observable by means of science, Archimedicity is not required by nature,
8lid leaving it behind would possibly allow the treatment of the above two concepts.
30 it appears on the one hand legitimate and on the other hand intriguing to study

Such number systems, as long as the above mentioned essential properties of the real

:Eers 9J9 PTM‘“

g, 1
here are :1.npte v
a P . ‘ u
4*: etererarnpi t

1.1 :rataueeper le
-4224. n "f"
z... fleiIaSUIl 2c] ]. I)
its store Cm rza o
frets.
.. . ;i.,
h nportnt .w

_.. 't-'

(

... whiz" we“ I
‘1. plfl'; ‘ - ~.‘
In. ‘cfud-LD mll\m(l:
:e and. perhaps run
;::.or tnost' of th
llr“:"‘M;l :
......0301 a gen

it. re
were to the

All“ Permits {[1

Palm
«namely. the
it} tartT ' I
._ large. it IS
Mei 0n the other
«i. an be represen-
.aoreonputation

‘2‘ '.
"~ of Far

”Wed '27

If'

3.x;r.‘ . .
“Dan 14‘: l
:7“; 1 ‘I
“iii-tr

 

numbers are preserved.

There are simple ways to construct non-Archimedean extensions of the real num-
bers (see for example the books of Rudin [36], Hewitt and Stromberg [19], or Stromberg
[44], or at a deeper level the works of Fuchs [16], Ebbinghaus et al. [15] or Lightstone
and Robinson [29]), but such extensions usually quickly fail to satisfy one or several
of the above criteria of a “useful” field, often already regarding the universal existence

of roots.

An important idea for the problem of the infinite came from Schmieden and Laug-
witz [37], which was then quickly applied to delta functions [21, 23] and distributions
[22]. Certain equivalence classes of sequences of real numbers become the new number
set, and, perhaps most interesting, logical statements are considered proved if they
hold for “most” of the elements of the sequences. This approach lends itself to the
introduction of a general scheme that allows the transfer of many properties of the
real numbers to the new structure. This method supplies an elegant tool that, in

particular, permits the determination of derivatives as differential quotients.

Unfortunately, the resulting structure has two shortcomings. On the one hand,
while very large, it is not a field; there are zero divisors, and the ring is not totally
ordered. On the other hand, the structure is already so large that individual numbers
can never be represented by only a finite amount of information and are thus out of
reach for computational problems. Robinson [34] recognized that the intuitive method
can be generalized [25] by a nonconstructive process based on model theory to obtain
a totally ordered field, and initiated the branch of Nonstandard Analysis. Some of the
Sta-nda.rd works describing this field are from Robinson [35], Stroyan and Luxemburg
I45], and Davis [14]. In this discipline, the transfer of theorems about real numbers is

eth'emely simple, although at the expense of a nonconstructive process invoking the

2:105 choice. Ieadi
7:5 :ntonstructiven
r ample. the tart

getire or negatite. c

iilihfl approach
if rte pitneered ht
{basins and tom

1. .. L 1 ._
-5: -13..le C3D Eh."

 

Kiss: How Two

3::5‘ 30. Ollie

1.2 Outline

273;: risertation. ii
iii the real ntunl
in: young Levi-I
i239": field that is C

Tl." 'yl m
“~44? ”1 -:
Demfmnté

*.
J

-. aerentiahle fur

11‘??? ; V

In.
1',“
'J

{tit appeared

J}
-_L;1"i.v_ r .
{W 24‘ T
a J.

;- L, .
“65:1"? . -
ind in the]
'3:an .
‘ ~11?»
vs
“5:21;?

tells a S] VIP

axiom of choice, leading to an exceedingly large structure of numbers and theorems.
The nonconstructiveness makes practical use difficult and leads to several oddities;
for example, the fact that the sign of certain elements, although assured to be either

positive or negative, can not be decided.

Another approach to a theory of infinitely small numbers originated in game theory
and was pioneered by John Conway in his marvel “On Numbers and Games” [13].
A humorous and totally nonstandard yet at the same time very insightful account of
these numbers can also be found in Donald Knuth’s mathematical novelette “Surreal
Numbers: How Two Ex-Students Turned to Pure Mathematics and Found Total

Happiness” [20]. Other important accounts on surreal numbers are by Alling [1] and

Gonshor [17].

1.2 Outline

In this dissertation, new elements of analysis on a different non-Archimedean exten-
sion of the real numbers are discussed. The numbers R were first discovered by the
brilliant young Levi-Civita [27, 28] who succeeded in showing that they form a totally
01‘ dered field that is Cauchy complete. He concluded by showing that any power series
With real coefficients converges for infinitely small arguments and used this to extend
r 83-1 differentiable functions to the field. His number system has subsequently been
rediscovered independently by a handful of people, including M. Berz [3, 5, 7], and
the subject appeared in the work of Ostrowski [32], Neder [30], and later in the work
of Laugwitz [24]. Two modern and rather complete accounts of Levi-Civita’s work
can be found in the book by Lightstone and Robinson [29], which ends with the proof
of Cauchy completeness, and in Laugwitz’s account on Levi-Civita’s work [26], which

E1180 contains a summary of properties of Levi-Civita fields.

ifhaettf 2' “e
1;: which Will 8’1
3; pore :ts structur
3: session of R h
p; ,3: rational 'lllll
re... 53): or I01 .
if: .25 R. which is re
:1 seer er present
; :r'p’V'ls I101“
2:321 number and
:1: he results llttj'E;
:eratrehirnedear,

Info pter 3. we

“t questions .-
1in'that R ad::.
““3339 lmag Ii.
31'3" tattered non-A.
1 Tittt impriugy i:
' 1;;9} Small pm;

Z- 138% top.
3?; tor the Study o
In” fl'JIiOWing cl.

. a. ‘
f '4. '
Nau'fls

011 the I.

:1“. :E'.

“e
of!“ Willi“ .'

r
”‘1‘ .53]: 9
i

in: .
he “93k to-

 

In Chapter 2, we prove general results about skeleton groups and field automor-
phisms, which will serve as an introduction to the field R and will help in understand-
ing more its structure reviewed in Chapter 3. We show that if a non-Archimedean
field extension of R has roots of positive elements, then its skeleton group must con-
tain the rational numbers Q and we show that the skeleton group of R is Q. This
already says something about the uniqueness of R as a non-Archimedean field exten-
sion of R, which is reviewed in Chapter 3 below. We show that every automorphism
on R is order preserving and that, contrary to the real case, there exist nontrivial field
automorphisms; however, we show that if P is an automorphism on R and if q is a
rational number and r a real number then P(q) = q and P(r) is approximately equal
to r. The results mentioned above are not unique to R and hold in the same way for

any non-Archimedean field extension of R which has roots of positive elements.

In Chapter 3, we review some of the work done by M. Berz in [3, 5, 7, 9]. We
begin with questions about the algebraic, order and topological structures of the field
and show that R admits nth roots of positive elements; more so, the field obtained by
al'djoining the imaginary unit is algebraically closed. It is shown that R is the smallest
t(“Sally ordered non-Archimedean field extension of R which is Cauchy complete in
the order topology, in which positive elements have roots and in which there exists

an infinitely small positive number d such that the sequence (d") is null in the order
toDology. A new topology, complementing the order topology, is introduced, which is

useful for the study of power series in Chapter 4.

In the following chapters, we extend the previous work and formulate new aspects
of analysis on the Levi-Civita field R. We start in Chapter 4 with a review of
ConVergence of sequences and series with respect to the order and weak topologies
Which leads to the proof that R is Cauchy complete in the order topolog while it

IS not in the weak topology [5, 7]. We prove new results on convergence; especially

555119an Will] ii
sci science a weak t
:12 and a deviled s'
.11 :31. the direct :1
. frictional do;

3:3. 10. 43...

‘4
1

("caper 5. m

E

lie: ret'iewits.
is is 11 any metr
fimaamm
2135:1011 of func:.
13.31 or rmrrirrg;
:.res ctteria 1.
tint-0113 or differs:
ififldsmm.

II51363 review 5:;

:13“? Of the dam.

W—i

5:*z. .

.11.,‘1 tile definition

3:“? :;,
N 01 strong Cfr‘

’1131'. . ,
{15.10}. “I? a
‘ mere result.

I'

17:»;
find around a .

 

those dealing with the sums and products of sequences and series. We then review
and enhance a weak convergence criterion for power series. This allows the extension
to R and a detailed study of all transcendental functions in Section 4.4. This also will
allow for the direct use of a large class of functions in Chapters 6 and 7, in particular
all the functional dependencies that can be formulated on a von Neuman computer
[38, 39, 40, 42].
In Chapter 5, we start by extending and generalizing the calculus developed in
[5, 7]. After reviewing topological continuity and topological differentiability, we show
that like in any metric space, the family of topologically continuous or differentiable
functions at a point or on a domain is closed under addition, multiplication and
composition of functions and that if the derivative exists, it must vanish at a local
maximum or minimum. Unlike in R, however, we show with examples that these
smoothness criteria are not strong enough to guarantee that a function topologically
continuous or differentiable on a closed interval of R satisfy the common theorems

0f real calculus or even be bounded.

We then review stronger definitions of continuity and differentiability based on the
coIlcept of the derivate [10]. We show that these are preserved under operations on
full(:tions; and we derive a chain rule and a tool for easily checking the differentiability
of filnctions on intervals of R. Also in this chapter, we generalize the central result in
[3’ 5 , 7, 10] that derivatives are differential quotients after all. This offers a pretty way
of doing computational differentiation; see Chapter 7 and [38, 39, 40, 42]. We then
reView the definition of high order differentiability, the remainder formulas and the
dotIlain of strong convergence of the Taylor series for infinitely often differentiable
furletions [10]. We also study weak convergence of the Taylor series and use that
to Drove more raeults about power series; in particular, we show that they can be

reex‘panded around any point of their domain of convergence. This entails that the

.... studied in I
:5 iferehtiahility rr.

. - ‘ .:H ‘
~ .;.:-15 which 11 .1. s

-o
..v

5:21:11 theorem.

line the proper
11:111. criteria .

1.3335011 72 that

‘ 1
saucy-f 9

s... .11. .hese expat

>41,

1.: feet-cities. \‘11

5: recs' calculus it;
lies theorem and
11's are finitely nf'
r5 are aga‘r‘. .

rivjl . ; \
“his 11 Chaps

‘ t
i '2‘“

«any. the com}.

\I
'l

"' 3940- 43‘. We ~

3:331:19» .
, K‘Uie to rig“:

. ‘ 1
‘ 4... .
‘ "’a‘sf’n

~0- .u‘..Gbl€ 0r DI

‘.
'u

.. .

“'6

we. even if the

hi.
.9.

...szzon of the met

1 . .
Ekeflllmn
--~-..11;.3al mEthotis

Q’f‘v'.
r“.

 

functions studied in Chapter 6 are translation invariant. Finally, based on the concept
of differentiability mentioned above, we present a detailed study of a large class of

functions which will be shown to satisfy an intermediate value theorem and an inverse
function theorem.

Using the properties of the weak convergence discussed in Chapter 4 and the
smoothness criteria discussed in Chapter 5, we study in Chapter 6 a large class of
functions on R that contains all the continuations of power series from R to R. We
show that these expandable functions [41, 43] are closed under composition and arith-
metic operations. We also show that for these functions, all the common theorems
from real calculus including the intermediate value theorem, the maximum theorem,
Rolle’s theorem and the mean value theorem hold. Moreover, the expandable func-

tions are infinitely often differentiable and integrable; and the derivative ftmctions of

any orders are again expandable functions.

Finally, in Chapter 7, we discuss one of the important applications of the field
R; namely, the computation of derivatives of functions representable on a computer
[38, 39, 40, 42]. We show that using the calculus on the non-Archimedean field R,
it is possible to rigorously decide whether a function representable on a computer
is differentiable or not at any given point, and if it is, to accurately determine its
deriVative, even if the coding exhibits nondifferentiable pieces. Details of an imple-
mentation of the method and examples for its use for typical pathological problems
are given. Execution times for both standard problems as well as exceptions where

Conventional methods fail are compared with those obtained using the conventional

algorithms.

lliflé section. we I
21.1 the intuit inj
i ticketed by the

“7;; gbtalsfl ll)? cl.

, .
.1. :e “SHE-fr as 8 I

in w-y ia; ‘ “ '
_ .1,“ the pcfl'tf‘ffi
"QM“; N'M‘P " l '
.osmuub umdllfn‘ tal(
r3:I.-.D *‘P. h
.11. .11. t zippers:
“ :.I‘: .1. .

. . 2 limit) Clinlll
’rdr'i '
hon-Archirr‘re
rec“.

«as a formed 1'
4"“; :I'o" 5' -

action group
,1: That rs. ere

r‘

s

. ..sgter 2. the she
BF?“ ' '
...se 01 tech:

in“! "
«seer. - .
. 1 1‘ cm ascent-

If};
“freer"
“wt ._ .
cpflC‘e Of'

 

1.3 Intuitive Remarks

In this section, we present general remarks about the Levi-Civita field R that will
enhance the intuitive understanding of the structure of the field discussed in Chapter
3. Motivated by the need for differentials for applications in Physics (see Section 1.4),
we will obtain the differential d in Definition 3.5 and show that every element a: of R
can be written as a formal power series of d

as: qud", (1.1)

960
in which the powers of d, also called the support points of :13, form a left-finite set of

rational numbers; that is, below any given rational number t only finitely many powers
smaller than t appear in the series in Equation (1.1). We remark here that Equation
(1.1) is directly connected to the Hahn theorem [18] which holds for a general totally
ordered non-Archimedean field F and which states that every element of F can be
Written as a formal power series in which the exponents form a well-ordered subset
0f the skeleton group Sp of the field F; see Chapter 2 for the definition of skeleton
groups, That is, every subset of that set of exponents has a minimum. As we will see

in Chapter 2, the skeleton group of R is SR = Q.
Because of left-finiteness, the set of rational powers in Equation (1.1) can be
arranged as an ascending divergent sequence, and we can rewrite Equation (1.1) as

00
a: = andq" with qj, < qu if jl < jg, (1.2)

n=1
Where convergence occurs with respect to the order topology of R. We remark here
that Equation (1.2) is proved directly in Chapter 3 without reference to the Hahn
theorem, but using the left-finiteness of the support points and the properties of the

order topology. John Conway also proved directly a similar result for his surreal

nu111bers [13].

The fact that th:
51 fer the positive
‘ ll . '
issgehrarca.) c.
1.15 15 to define 1:.
7:31 the cones] ,.
iii 'the rations-1i
‘5 "; giien ratios..-
3292 Tea. and 3;

3:21:23: 36. we 11..

1:311 of the left-ti
site for each q.
:;l~::.e3:atior1 of the

.513115 threctiy f

1.4 Applica

t
1;,

.. 1111c mention:-

1‘11an in R 21

2111-1 61.62 : R *1

651‘] z I

and

1
67,11) I

II

 

4.1315316) (hit

4
-r,

The fact that the powers of d are rational numbers, rather than integers, is neces-
sary for the positive elements of R to have roots in R; and this is used to show that
R is algebraically closed [5]. The left-finiteness of the supports of the R numbers
allows us to define multiplication of two given numbers :1: and y by multiplying, term
by term, the corresponding rational power series of a: and y and obtain a new element
of R. In the rational power series representation of the product, the coefficient of d"
for any given rational number q is obtained as the sum of finitely many terms: Let
a: = Esq xrd' and y = ZseQ y,d‘ be given in R, and let 2 == :1: - y; then according to
Definition 3.6, we have that z = quQ 2qu where for each q

2,, = Z wrys. (1.3)

r+s=q
Because of the left-finiteness of the supports of a: and y, the sum in Equation (1.3)
is finite for each q. Moreover, the left-finiteness is a necessary condition for the
implementation of these numbers on a computer as we will discuss in Section 1.5; this

also follows directly from the Hahn theorem [18].

1.4 Applications in Physics

AS We have mentioned in Section 1.1, the existence of infinitely small and infinitely
large numbers in R allows us to have well-behaved delta functions; for example the

functions 61, 62 : R —+ R, given by

_ d'1 if |x| 3 d/2
5‘9"”) ‘ { o if |x| > d/2
and
6 _ 0 if Ix] is infinitely larger than d
2(3) 71;;exp(—x2/d2) otherwise ’

Whel‘e d is again the diflerential introduced in Definition 3.5, are piecewise expandable

(see Chapter 6) delta functions; they both assume infinitely large values at 0, they

.25"; at all other
it: we can replace
tiara E R‘. 81;.
5; Thus. each of t':

1,3: rnttttns.

 

 

 

l1: major motiv
..:R to the con
L::1:1rders. l'siflfi’. 1
1mm and s
21:} $1111 rea‘ p1
Site: 11 realxalxwi
3:75;;etit1n throng
:11: a real point 1

.:ezrepresezrtirre j
L.

h .
_V~-w,9.

wastes of f at r
hfeger m. f 1
21:13:31“ if there 0:

‘6, 911d flr 2..

:W;-"ll P'A
7‘ M3919 c

10

vanish at all other real points and their integrals are equal to one. We also note
that we can replace d in the definitions of 61 and 62 with any other infinitely small
number a E R+, and obtain delta functions that behave in a similar way as 61 and
62. Thus, each of the two delta functions defined here generates a whole family of

delta functions.

The major motivation for this work is the application of the existence of differen—
tials in R to the computation of derivatives of complicated real functions up to very
high orders. Using the calculus on R developed in Chapter 5, we derive in Chapter
7 a necessary and sufficient condition for the derivatives of real functions to exist
at any given real point and show how to find the derivatives whenever they exist.
Given a real-valued function f that is obtained from the intrinsic functions and the
step function through a finite number of arithmetic operations and compositions, and
given a real point r, we show that we can extend f to R and define it at r :l: d.
Then representing f (r :l: d) as expansions in powers of d allows us to isolate the (real)
derivatives of f at r as coefficients in the expansions. We show that for a given pos-

itive integer m, f is m-times differentiable (in the real sense) at the real point r if
and only if there exist real numbers 011, . . . ,am such that, up to the power m of d,

f (7‘ ~ d) and f (r + d) are given by

f(7"" d) =m f(7“) + :(—1)jajdj and
f(r + d) =m f(r) + :ajdj, (1.4)

in Which case the derivatives of f at r are given by

f(j)(r) = jlaj for allj E {1,-",m}'

Remark 1.1 Another necessary and sufi‘icient condition for f to be m-times dif-

f"5"‘°5372.tiable (again in the real sense) at r is that for all n E {1,...,m}, we have

   

1.7.5.. 7;; 111,05: fimte t

' "f 11
31:12:69 are 911. -

{gutter "1.4) or R,
item at a real p.
tease d L5 infinitelj

LL. se:tiort are in

he ability to Ca
313.13g. in Bea:
Erie. to be able to 11
i”... the llallSCE‘ll‘
.iiszezte‘rl in C OSY
7 TDSY INFINITY
51$: orders ere

2113301 affect t3-

«1.11M, .

Auvrentlar ;'

1‘: e 7 ,

..h
I f. n
‘l‘i

(‘1‘ ll‘G (l .
‘ 4~ xerrvat

 

11

that d'" (z?=o(_1)n-j ( 7; ) f (r +jd)) and d‘" (ELM—1V ( 7; ) f (r - 311)) are
both at most finite in absolute value, and their real parts agree. In this case, the real

derivatives are given by the respective real parts of the diflerence quotients. That is,

l=o(—1)"_j ( 2 ) f(7‘ +jd)
d"

 

Me) = a

for all n E {1, . . . ,m}. In particular, if f is differentiable at r, then

W) = ,R (fir + dc); fed).

 

Equation (1.4) or Remark 1.1 allow us to compute the real derivatives of a real-valued
function at a real point to full machine precision and with no numerical penalties
because d is infinitely small. The proof of Remark 1.1 and of all the statements made

in this section are found in Chapter 7.

The ability to calculate derivatives to high orders is important in many areas of
PhYSics, e.g. in Beam Physics [9], non-linear Dynamics and Celestial Mechanics. In
Order to be able to use the theory developed in Chapter 7, the arithmetic operations
and all the transcendental functions which are extended to R in Section 4.4 were im-
Plemented in COSY INFINITY [6, 8, 12]. Because of the theoretical work in Chapter
7: COSY INFINITY can now compute derivatives of very complicated functions to
very high orders even if the coding contains if-else or other nondifferentiable pieces
that do not affect the final result. Formula manipulators such as Mathematica, Au-
tomatic Differentiation methods [11] and even previous versions of COSY INFINITY
[4] Were not able to handle such cases Even when Mathematica works, our method
is Inuch faster since no symbolic differentiation is required before the numerical eval-

nation of the derivatives. Moreover, the results obtained are accurate up to machine

.Lv' ~su'

2.; E: of 82." PW"

Rep,” tieal 1

I O‘kzn, : H
;" 1.5-Ill .Ol all pl
7. PK PP” '
._ anvil-m1
the real ea
.;:-.’5‘e '7
.,1.-t5 . thro'ts
.1." ”l V" I
.al’t 10111111522

.: :temeeliate \

«1.3115 that do 1

, .
‘ P
. , ,

.‘l
. 1“ l "
‘ RI gl\‘_‘

lllflflfiqjm'

f" '
-14.“ij l
lhf‘mm
1
.JH [0 '1 l
.. ‘ l
\ r '

12

precision; this represents a clear advantage over traditional numerical differentiation
methods in which case finite errors result from digit cancellation in the floating point
representation and for high orders the errors usually become too large for the results

to be of any practical use.

The practical usefulness of the existence of differentials is obtained at the cost that
the field is disconnected in the order topology. The disconnection occurs because
if a: and y are any two positive elements of R and if x is infinitely smaller than
y, then for all positive integers n we still have that n2: is infinitely smaller than
y. This disconnection between the different orders of magnitude makes it hard to
extend the real calculus to R and explains the need for all the mathematical work in
Chapters 2 through 6 before we get to the applications in Chapter 7. In particular,
there are topologically continuous functions on closed intervals that do not satisfy
the intermediate value theorem, the maximum theorem, or have multiple primitive

functions that do not differ by a constant. For example consider the simple function

f1 [0,1] —-> R, given by

0 if a: is infinitel small
f(x) ={ y .

1 if a: is finite

Then f is topologically continuous and differentiable on [0, 1], but f does not assume
the Value d on [0, 1] even though f (O) < d < f (1). Moreover, even though f’ (as) = O
for all a: E [0, 1], f is not constant on [0, 1]. This is due to the fact that the behavior
0f f in the infinitely small part of the domain, where it is constant and equal to zero,
is totally disconnected from its behavior in the finite part of the domain, where it is

constant and equal to 1. More examples are found at the end of Section 5.1.

The difficulties mentioned in the previous paragraph are not specific to R and
are common to all non-Archimedean structures, which explains the previously rather

limited results that could be derived in N on-Archimedean Analysis. In Chapter 5

(“Hater 61 “1"

13‘1“

:1 356 {Sal 10g?“
.l- n" "l' l “1:11
‘15 fistula LL“
$121315 of cont 1:
: Lemar-ate 1:1
33:29: 11e33itlozz:
.at a. the com:
.13: are [men 1111
... tier sat‘stt' t.
1;: rattrerrrents o:
H,” _ l
battlef- Llel' sat

tater: and they

1.5 Impler

If.“ -‘ ~ ‘~“
" r] , -:7- a I
J¢\.¢:\ Ll) All lll‘
0 ac

21.131239] can be

371:: they differ

.L

. 4,, 1, ,,
wpfllt’flidi

“" '
. 1., 9‘ SlIleeR

'9

ssrttttre in a \‘t

1:? 91'? .
I} bound,

":3: T l
1 the left of

‘3‘: (:9. d
“ - Earth I

i
3..
Kai”?

..‘\\ r

1 t
J:

13

and Chapter 6, we provide elegant solutions to the problems mentioned above; and
we use that together with the results developed in the previous chapters to show
the practical usefulness of the Levi-Civita field in Chapter 7. We first enhance the
definitions of continuity and differentiability in Section 5.2 and obtain in Section 5.4
an intermediate value theorem and an inverse function theorem based on the new
stronger definitions of continuity and differentiability. We then show in Chapter 6
that all the common theorems of real calculus hold for a large class of functions
which are given locally by power series with R coefficients. In particular, we show
that they satisfy the intermediate value theorem even though they do not satisfy
the requirements of the general intermediate value theorem discussed in Section 5.4.
Moreover, they satisfy the maximum theorem, Rolle’s theorem and the mean value

theorem; and they are infinitely often differentiable and integrable.

1 .5 Implementation

Besides allowing illuminating theoretical conclusions, the strength of the R numbers
is that they can be used in practice, and even in a computer environment. In this
respect, they differ from the nonconstructive structures in Nonstandard Analysis [25,

35].

An implementation of the R numbers is not as direct as one of the Differential A1-
gebras [2, 9] since R is infinite dimensional. However, it is still possible to implement
the structure in a very useful way. Since there are only finitely many support points
below every bound, it is possible to pick any such bound and store all the support
points to the left of it together with the respective coefficients of the corresponding
powers of d. Each R number is represented by its support points, the respective co-

efficients of the powers of d, and finally the value of the upper bound of the support

.48; 'I'haI 15.1.
[.u'r

, ‘ A ’1- "hill
.1: {IL-9‘ ”Mm“

' ' .‘7 ~
..q-f it the all“

tier: if is the ll

tested. In p

EMW‘

Ft: 1111 of W
11.3.1111 of tl

hand of ti;

L“ r
J» 1w h

; ,la . .‘
«.5. Will“ “I

31:. 1:15 the two

El“*\'~ . ' '
\V*« b Cdfllfffl

14

points. That is, if we limit ourselves to n terms in the expansion, we can represent
the series expansion of a given ’R. number a: by n pairs of numbers, the first n powers
of d in the expansion and the n corresponding coefficients in the following way
$1 $2 . o u :12"
$ =
M {,1 q. ,1}

=M and“ +3320”2 + ' ° ' + mndq",

where M is the upper bound below which all the support points q1,q2, - ' - , qn of :1: are

to be stored. In particular, a real number 1' will be represented as follows

Wt}

=M Tdo

if M 2 0 and r + d is represented as follows

T+d 2M {6 i}

=M Tdo +1d1

ifM_>_l.

The sum of two such numbers can then be computed for all values to the left of
the minimum of the two upper bounds; so the minimum of the upper bounds is the
upper bound of the support points of the sum. In a similar way it is possible to find
a bound below which the product of two such numbers can be computed from the
bounds of the two numbers. Altogether, the bound to which each individual number

is known is carried along through all arithmetic.

For the purpose of the implementation of the elementary functions, we make use
of the addition theorems, e.g. Theorem 4.13 and Theorem 4.14, proved in Section 4.4
to truncate the series at a certain depth M. The elementary functions are defined

for any number a: E ’R. that is at most finite in absolute value. Any such a: can be

me: as I = T
.. . r , .
.135»: lflf‘Ult‘l

h

reared pate

15 zit: pasztzx‘
I 1 l

:5 "rm-cu com

‘ 'H .
73"". ' “'
Juli-.3516: 5L ..Lc‘t ['

.. ‘
;v~~ a. mr.‘ ~
...rl.’ Vupmfi‘

Example 1.1 l

ample 1.2 L

”2
{1'
H4‘

15

written as a: = r + s, where r = §R(a:) and where Isl is infinitely small. Thus using the
addition theorems for the elementary functions, we separate the function value at :1:
into a real part and an infinitely small part which can be represented by a power series
in s with positive exponents. This power series in s can be rewritten as a power series
in d which converges fast and is truncated at the desired depth M. The following
examples illustrate how the truncation is done for the division and the sine function;

similar schemes are followed for the other transcendental functions.

Example 1.1 Let a: = r + d, where r is real and let M = 10.

 

Then
1 _ 1
a: _ r+d
00 j
rj___0 r
l 1 12 l 10
:10 ;—-T—2d+'73d —°"+m‘d .

Example 1.2 Let :c = r + d, where r is real and let M = 10.

Then

sin(x) = sin(r+d)

= sin(r) cos(d) + cos(r) sin(d)

_ , 1 1 1 1 3 1 10
—10 Sln(r)(1_§ldz+4—!d4—6ld6+§ld _Tf)—!d )
1 3 15 1 7 l
+cos(r) (d—é—ld +B-!d _7fd +§d9)

= sin(r) + cos(r)d — iii—Edd? — £3532? + ' - - — Lfiggdm.

1f

.. ”y“
<rz. .
3:19:11 “La

 

. n. N“ r F;
.r ,
. Skim“
.. .

’ I
f' I;’ “17.59
I. 1. Lin;

'h.,' ‘
. A _‘
’ulfi! l“ p.-
. \r'\:
A
Is ‘ _ ‘
'T’ 5’3"“?

3.: 75 Milk.

’ A
k'i.

,L ‘M‘
s» \

“-~. m ‘
“.9.“ ~' ,
x: \ ”1?) t,

-..H .1

rug
'k
.U
.‘_

, ltfhfld (

L,
“mile 1.

16

The real functions sin and cos already exist on the computer; and since the power

series expansions of cos(d) and sin(d) converge very fast for the infinitely small d, the
truncation is done easily as shown in the example above. Similar arguments hold for

the other transcendental functions.

Using the rules outlined above, all the arithmetic operations and all the transcen-
dental functions have been implemented in COSY INFINITY [6, 8, 12]. This allows
us to apply the theoretical results of Chapter 7 for the computation of derivatives
of real functions, in which case the upper bound of the support points in the final
result must be greater than or equal to the order m of differentiability we would like
to check. This final upper bound is calculated in terms of the upper bounds of the
intermediate calculations, which we may have to change to get the desired final up-
per bound of the support points and hence obtain the desired information up to that
depth. The number of support points of f (r + d) or f (r — d) smaller than or equal to
m may depend on the complexity of the function f; however, using Equation (1.4),
we can decide the differentiability of f at r up to order m and obtain the derivatives
up to machine precision just by looking at the first m + 1 support points of f (r — d),
the first m + 1 support points of f (r + d) and the corresponding coefficients. We give
three simple examples here and refer the reader to the examples and the details of

the method discussed in Chapter 7.
Example 1.3 Let f1 : [—1, 1] —> R be given by f1(a:) = exp(:c)

Then evaluating f1(:l:d) up to the power m of d, where m is a positive integer, yields

f1(-d) =m 1+Z(-1)j%dj and
i=1 '

Example 1:

.4”.

Iflpyto bed!

I... 4‘54.

n, .

I
H.

hample 1

17
Since f1(0) = 1, we obtain that fl is m-times differentiable at O with derivatives
(1') _ 1 '
f (O)—3;forallg_<_m.
Example 1.4 Let f2 : [—1, 1] —’ R be given by
f2(x) = |:1:|7/2 sin(xl

Then, evaluating f (id) up to the power 8 of (1 gives

 

13/2

f2(-d) =3 -d9/2-d6 and
13/2

f2(d) =8 d9/2+d6 '

Since f2(0) = O, we obtain using Equation (1.4) that f2 is four-times differentiable at

0 with derivatives equal to 0; but f2 is not m-times differentiable at O for any m 2 5.

Example 1.5 Let f3 : [—1, 1] —-) R be given by

v.) ={ gel-“W :5: :3.

Then, evaluating f3(:l:d) up to the power 2 of d yields
f3(id) =2 0 = f(O),

from which we obtain that f3 is twice differentiable at 0 with derivatives
fé(0) = 2521(0) = 0.

To check whether f3 is three-times differentiable at O, we need to evaluate f3(:td) up

to the power 3 of d; if we do so, we obtain that

f3(-d) =3 45/2

f3(d) =3 (15/2,

I v

" " ‘Ff ”keys
2 Lin inf: §6Al4

-0

18

from which we infer, using Equation (1.4), that f3 is not three-times differentiable at
0 since 5/ 2 is a noninteger number smaller than 3 and the coefficient of d5/2 in the
expansion of f3(d) is not zero. The same conclusion follows also from Remark 1.1,

since the difference quotient of order 3,

f3(3d) “ 3f3(2d) + 3f3(d) " f3(0)
d3 ’

is of the same order of magnitude as (1-1/2 and hence it is infinitely large in absolute

value.

1.6 Notations

Throughout this dissertation, we will adopt the following notations:

Z, Z +, Z ‘2 the set of all integers, the set of positive integers and the set of negative

integers, respectively;

Q, Q+, Q‘: the field of rational numbers, the set of positive rational numbers and

the set of negative rational numbers, respectively;

R, R+, R‘: the field of real numbers, the set of positive real numbers and the set

of negative real numbers, respectively;
L: the field of the formal Laurent series; and

7?. (read R—script): the Levi-Civita field.

Chapte

Skeleto
Automt

Ltzschapter.‘
“Ax :‘ ‘Llfi', "
..1; “Mil WI

23444 .“‘.
h ““‘l “Lil be

.:F,
.-.hatmvf

”lfimdonv

.§.Lc,“r, ‘

Chapter 2

Skeleton Groups and Field
Automorphisms

In this chapter, we prove general results about skeleton groups and field automor-
phisms which will be useful for understanding the structure of the Levi-Civita field

R, which will be introduced in Chapter 3.

2.1 Skeleton Groups

Let F be a totally ordered field, and let a, b E F " = F \ {0} be given. We say that
a N b if and only if there exist n,m 6 2"” such that nlal > |b| and m|b| > |a|, where

I . | is the usual absolute value on F, defined by

l$|_ a: ifoO
_ -:r if$<0'

Then ~ is an equivalence relation.
Let Sp denote the set of all equivalence classes. Then
Sp = {[a] : a E F‘}.
Let a,a1,b,b1 6 F‘ be such that a ~ a1 and b ~ b1. Then a . b ~ a1 - b1. Define

@ISFXSFHSFby

[0169113] = [a-bl-
19

s": it ‘5 east '
. -‘ l r - ' .
agent I: 0mm

of F. Tl

r v
gen-air“

A.‘ f
when =

3
’h

u
’ . I .
T . ‘, -
‘Jth ~36 relatll‘lz‘

'IV" .1 F‘ .'
'il‘atrfi‘ dSeLlLlIl g

I

‘.’. .1‘ -
di'deL al.]

V

and 5

Sa

1:.lht]‘~ ,
eAlerua":

' ‘1, {,3 .
a. Jr: id («"71“
. H ‘5-

, .u.
) F ‘n

l
“ l“ 0‘3 refm.

.V
U‘
2‘55 ,3
«i ”p 4‘. r
"d, \ '
~I‘E‘et0r]
‘
5‘
.‘ .

20

Then it is easy to verify that (Sp,€B) is an abelian group whose additive neutral
element is denoted by 0 and is given by O = [1], where 1 is the multiplicative neutral
element of F. The additive inverse of an element [a] E S F is denoted by e[a] and is
given by GM = [a'l].
Let a,a1,b,b1 E F“ be such that a ~ a1 and b ~ b1. Assume that, for all n 6 2+,

n|a| < [b]. Then ”[01] < [b1] for all n E Z+. Define <, 5: SF —+ 5,: by

[a] < [b] if and only if n- |a| < |b| for all n E Z+

[a] S [b] if and only if [a] = [b] or [a] < [b].
Then the relation 5 defines a total ordering on (S F, GE). Thus (S F, GB, 3) is a totally

ordered abelian group; that is,
1. for all [a], [b] E (Sp, 69), [a] S [b] or [b] S [a], and [a] = [b] if and only if [a] _<_ [b]
and [b] S [a],
2. for all [a], [b], [c] E Sp, [(1] g [b] => [a] EB [c] S [b] EB [c].
In the following, the totally ordered group (S 1:, EB, S) will be simply denoted by S p
and will be referred to as the skeleton group of F.

One can easily verify that the skeleton group of R is given by

SR = {0} = {[1]};
and the skeleton group of L is given by
S1, = Z,

where L is the field of the formal Laurent series. After having introduced the totally
ordered field 1?. in Section 3.1 and Section 3.3, one can also verify that the skeleton
group of ’R. is

S1; = Q.

Definition 2.1 l

2" :13 on” If the
Theorem 2.1 L-(

Proof. Since F
135151: Sin
313161931th of

' .‘ ,; A] . 1
3: Liam sum.

L?» that

(“ink-in; d

I}? , '
\ RM
""491 Dr ,
I ‘98P“.

’L’

(r7 I

hi; .
k‘jfihf’v

. , ,7.

21

Definition 2.1 Let F be a totally ordered field. Then we say that F is non-Archimedean

if and only if the skeleton group SF of F contains more than one element.
Theorem 2.1 Let F be a totally ordered non-Archimedean field. Then Z C Sp.

Proof. Since F is non-Archimedean, there exists an element d 6 F “‘ such that
[d] 73 0 = [1]. Since [d] = [—d], we may assume that d > 0, where 0 is the additive
neutral element of F. Since e[d] = [d‘l] E Sp, we may assume that [d] < 0, i.e. that
d is infinitely small. Consider the subset Z p = {[d"] : n E Z} of Sp. For m > n, we
have that

e[d"]€B[d'"] = [d‘"-d'"]=[d'"'"]=[d-d-...-d]

(m-n) times

=1d]@[d]i.i...@[dl

(m—n) times

 

<1d1old1g~old1
(m-n-l) times
< [d] < 0.

Thus, m 79 n => [d"‘] 75 [d"] for all m,n E Z. The map P: Zp —> Z given by
P([d"l) = -n
is an order preserving isomorphism; that is,

1. P is bijective,

3, P5 mmpatf

and

3. P is rompati

.22.: Z 5 sonny.»

Theorem 2.2 16‘

i
b

:f:-:.~':t:t‘€ element

Proof. Since F is 1
5.90th15;

:1 f“ < 0. [31:

1.31“] In pa

Now let q: 7; q;

L“: tense

22

2. P is compatible with the groups’ operations, i.e. for all m, n E Z,
P([dm169 W1) = P([de + P([d"l),
and

3. P is compatible with the groups’ order relations, i.e. for all m, n E Z,

W] < W] is P(ldml) < P([d"])-
Thus Z is isomorphic to a subset of Sp, or simply Z C Sp.

Theorem 2.2 Let F be a totally ordered non-Archimedean field which admits roots

of positive elements. Then Q C Sp.

Proof. Since F is non-Archimedean, there exists an element d E F * such that [d] < 0.
Let g > 0 in Q be given; write q = m/ n where m, n E Z +. As in the proof of Theorem

2.1, [d'"] < 0. Using the fact that [d’"] = [d""’] = [dq] EB [dq] EB - . - 69 [d9], we obtain
\_—/

n times

that [dq] < 0. In particular, [d9] 7E 0.

Now let ql 7E Q2 be given in Q. We may assume that (12 > ql. Then ()2 — ql > 0,

and hence

e[d‘“] e [(192] = [d‘n—QI] < 0.
Thus, ql aé q2 => [dm] # [d‘n].

Let Qp = {[dq] : q E Q}. Then Qp is a subgroup of Sp, and the map P: Qp -—e Q,
given by
P([qu = -q, (2.1)

is an order preserving group isomorphism from Q p onto Q.

2.2 Field .

lefinirion 2.2 tr

:Ltmof‘ptiem r

lemma 2.1 Let 5

.‘i'h‘l n, .
. mt mime I

Proof. Since P is

'\
-:"'
V"
\V
y' X-
r
x
t‘ ‘
~<Pfi .

23
Remark 2.]. Define ® I QF X QF —> QF' by
[(1%] ® [(192] = [6101-92].

Then (Qp, 69, ®, S) is a totally ordered field, and the map P given in Equation (2.1)

becomes a field isomorphism of Qp onto Q.

2.2 Field Automorphisms

Definition 2.2 Let F be a set, and let P : F —> F be given. Then we say that P is

an automorphism on F if and only if P is an isomorphism from F onto itself.

Lemma 2.1 Let S and T be fields, and let P : S —> T be a field isomorphism. Then

P has an inverse P’1 : T —-e S which is itself a field isomorphism from T onto S.

Proof. Since P is bijective, P”1 exists and it is bijective. Let +5 and +T denote
the addition operations in S and T, respectively; and let x3 and xT denote the
operations of multiplication in S and T, respectively. Now let yhyg E T be given,
and let 2:1 = P‘1(y1) and x2 = P‘1(y2). Then
P-1(y1 +7" 312) = 13—1 (P(331) +T 13052))
= P“1 (P(xl +3 232)) since P is an isomorphism

= (1314-3172 = P_l(y1) +8 FAQ/2),
and

P_1(y1xTy2) = P_1(P($1) XTP($2))
= P‘1 (P(xl x s 272)) since P is an isomorphism

= $1><s$2 = P_l(y1) XS P‘1(y2).

Thus P‘1 is a field isomorphism from T onto S.

Theorerr

~an '
7“; ‘
b A, .

Theorem

8
I. r‘fi- 2'7?
e ‘{ ‘ ‘, ‘

Proof. 3

24

Theorem 2.3 Let S and T be totally ordered fields, and let P : S —> T be an order
preserving field isomorphism. Then P”1 is an order preserving field isomorphism

from T onto S.

Proof. Using Lemma 2.1, it remains to show that P‘1 : T —+ S is order preserving.
So let y1,y2 E T be such that yl Sr yg, and let r1 = P‘1(y1) and x2 = P‘1(y2). We
need to show that 9:1 _<_3 r2. Suppose not; then .132 <3 9:1. Since P is order preserving,

we obtain that y2 = P(xg) <T P(xl) = y1, a contradiction.

Theorem 2.4 Let F be a totally ordered field, and let P : F —> F be an automor-

phism on F. Then P(q) = q for all q E Q.

Proof. Since F is a totally ordered field, Q C F. For any a: E F, we have that

P(x) = P(O + m) = P(O) + P(x); and hence
P(O) = 0.
Also for any :1: E F, we have that P(x) = P(l -:1:) = P(l) - P(x); and hence

P(l) = 1. (2.2)

Now let q > O in Q be given; write q = m/n where m,n 6 2+. Then m = n - q.
Thus
P(m) = P(n)-P(r1)- (2.3)

Using Equation (2.2), we obtain that for all I E Z +,

P(l) = P(1+1+---+L)
(times

P(l)+P(1)+---+(1]

1 times

 

1h.

1‘

A \ s

25

=1+1+~~~+1
ztimes

Thus, P(m) = m and P(n) = n. Substituting into Equation (2.3), we obtain that
P(q) = q. Finally, let q < 0 in Q be given; then —q > 0. Thus P(—q) = —q. Since
0 = P(O) = P(-q + q) = P(-—q) + P(q), we have that P(q) = —P(—q) = q. Hence

P(q) = q for all q E Q.

Corollary 2.1 The identity map I : Q —1 Q is the only field automorphism on Q.

Theorem 2.5 Let F be a totally ordered Archimedean field. Then the identity map

I : F —» F is the only order preserving field automorphism on F.

Proof. Assume not. Then there exists a nontrivial order preserving field auto—
morphism P on F. Thus there exists a: E F \ Q such that P(x) 76 2:. Since
P(—a:) = —P(a:) 7é —:r, we may assume without loss of generality that a: > 0.
Since P(x‘l) = (P(ac))"1 76 117—1, we may assume that x > 1.

By Theorem 2.3, P"1 is also an order preserving field automorphism on F, and
P‘1(x) yé P"1(P(:1:)) = :c. If P(m) < or, then a: = P'1(P(:r)) < P‘1(:r). So we may

assume without loss of generality that

1<:1:< P(m).

Since F is Archimedean, there exist n, N E Z+ such that

1
O<P(x)_x<Nandn_Nx<n+1

Then
n+1 S Nx+1< Nm+N(P(:c) —:1:) = NP(:1:).

1.. "'"f
.‘u‘tNa-a';

“1»- “To,
| .

..." \ aux i

7‘Kfi"'ir.‘r f

l h
'4 .ug A
v

lemma 2.:

. ' '
...l r] k
'16 1:, L“.

.‘m.
; If G: F
. -‘. .
"-.r
[.3 t I C
U

26
Thus,
Na: < n+1< NP($),

and since N > 0, we finally obtain that

n+1
N

 

x < < P(m). (2.4)

Applying P to the first part of Equation (2.4), we obtain that

 

n+1 n+1
P($)<P( N )‘T’

which contradicts Equation (2.4) itself. Hence the identity map is the only order

preserving field automorphism on F.

Lemma 2.2 Let F be a totally ordered field which admits roots of positive elements,

and let P be a field automorphism on F. Then P is order preserving.

Proof. It suffices to show that P(a) > 0 for all a > O in F; so let a > 0 in F be

given. Let b > 0 in F be such that b2 = a. Hence
P(a) = 13(1):!) = (19(5))2 2 0-

Since a 74 O and since P is one to one on F, we obtain that P(a) 75 0. Thus, P(a) > O.

Combining the results of Theorem 2.5 and Lemma 2.2, we obtain the following

result.
Corollary 2.2 The identity map is the only field automorphism on R.

Theorem 2.6 Let F be a totally ordered non-Archimedean field, let Sp be the skeleton
group of F, and let P be an order preserving field automorphism on F. Then the
map I‘ : Sp —> Sp, given by l"([a:]) = [P(m)], is a well-defined order preserving group

automorphism on Sp.

‘ T
,yuvk

I
. -,,.
. 3.1‘.‘

*:-1

S‘ rq.‘
‘u m1 T'
......J J.

\ .
3' ;..
“A
‘7’ .
a 1”
l2‘\ 9v
'.~’l'~
or“
d I
‘J I ‘I
~
\‘A
-‘l a. ‘
.. “Ell
F,
«(1"7.

27

Proof. Let a: e F“ be given; then P(m) yé 0. If x < 0, then P(m) < 0 and hence
P(lzl) = P(—:c) = -P(x) = ]P(x)]. On the other hand, if 0 < x, then 0 < P(m) and
hence P([x]) = P(x) = |P(:c)]. So for all 2: E F“, P(]:1:]) = |P(:z:)].

To show that f‘ is a well-defined map, we need to show that
[33] = [31] => [Pm] = ]P(y)]-
So let 23,3; 6 F ‘ be such that [:r] = [y]. Then there exist m,n E Z+ such that
|y| < m- ]x] and [51:] < n~ |y]. Thus,

|P(y)| = P(lyl) < P(m- [112]) = P(m) - P(lfvl) = W |P(Iv)|

and |P(:v)| < n- |P(y)|- He [P(en = [Po/)1.

Now we show that I‘ is one to one. So let [2:], [y] E Sp be such that P([x]) = I‘([y]).
We need to show that [as] = [y]. Since [P(m)] = [P(y)], there exist k,l E Z + such that
]P(y)] < k- ]P(x)] and ]P(x)] < l- ]P(y)]. Horn ]P(y)] < k- ]P(zr)] we obtain that

P(lyI) < P(k)-P(]:1:]) = P(k- lxl); and hence lyl < 16' l-Tl-

Similarly, ]P(x)| < l- ]P(y)] entails that [3:] < l - |y]. Hence [:12] = [y].

To show that I‘ is surjective, let [y] E Sp be given. We need to find [:13] E Sp such
that [y] = 1"([:1:]). Since P is surjective, there exists a: E F such that y = P(m). Since
y # 0, we have also that a: 75 0. Hence

[x] 6 SF and I‘([:1:]) = [PW] = [v]-

For any [x], [y] E Sp, we have that

I‘(Tml EB [31]) = P([3 - vi) = [P(33 - 31)]
= [P(m) - P(n)] = [P(m)] EB [P(y)l

= I“(11101) EB I‘(Tr/l)

11:63:31“ t

I l I ’
1. 1
r ‘ l
V\ I! <
15¢
I I ' ‘

Corollary 2.3

Corollary 2.4
1?? be on orde r

LE,D’T‘_i-\.]_

Proof. let r E

r .
‘ I
Hg. D . '
up“ ‘ 1'] — P
‘
. ‘ Ii
.. .

forollaq- 2.5

HP- :PiIVTi :

P

W B." The.

if: WWW; {1s
,

A“";4L l. m (v

_ map 0]]

28

It remains to show that I‘ preserves order in Sp. So let [11:], [y] E Sp be such that
[:13] S [y]; we need to show that I‘([:c]) _<_ F([y]). If [11:] = [y], then I‘([a:]) = I‘([y]).

Suppose [:13] < [y]; then for all n E Z+, n- |:r] < ]y]. It follows that
71' |P($)| = P(n) ° P(IJII) = P(n' ll‘l) < P(lyl) = |P(y)| for all n E 2+-

Thus [PM] < [Pu/)1; i.e- I‘([:c]) < P([yD-

Corollary 2.3 Let F, P, and I‘ be as in Theorem 2.6. Define A : Sp —+ Sp by
A([a:]) = [P'1(:r)]. Then A is an order preserving group automorphism on Sp, and

A=I“1.

Corollary 2.4 Let F be a totally ordered non-Archimedean field extension of R, and
let P be an order preserving field automorphism on F. Then, for all r E R‘, [P(r)] = O
i.e P(r) ~ 1.

Proof. Let r e R" be given; then [r] = [1] = 0. Hence, by Theorem 2.6, we have
that [P(r)] = [P(l)] = [1] = 0.

Corollary 2.5 Let P be an order preserving field automorphism on L. Then, for all
:1: E L‘, [P(m)] = [2:] i.e. P(r) ~ :r.

Proof. By Theorem 2.6, the map I‘ : SL = Z —> Z, given by I‘([:1:]) = [P(r)], is an
order preserving group automorphism on Z = S L. We need to show that I‘ = I, the

identity map on Z. By Theorem 2.6, we have that

[9?] = [21] => [P(m)] = ]P(y)] and [P‘l($)l = [P‘1(y)] (2-5)

[2:] < [11] => [P(m)] < ]P(y)] and [P’1($)l < [P‘1(y)l- (2-6)

We have that
1“(0) = 1“([11) = [P(l)] = [1] = 0- (2-7)

3.. l 9'
Mi n
.5: 1. we

in let

29

Let d be the L number representing the formal Laurent series :13. Since [(1] = —1 <

0 = [1], we have by Equation (2.6) that
[P(d)l < [P(1)T=[1l= 0 and [P_1(d)l < [134(1)] = [1] = 0-

Since d > 0, we have that P(d) > 0 and P‘1(d) > o. 11 P(d) = 11, then [P(d)] = [d].
If d < P(d), then [d] _<_ [P(d)], and hence —1 _<_ [P(d)] < 0. Since [P(d)] is an integer,
[P(d)] = —1 = [d]. If P(d) < at, then 11 < P'1(d). Thus —1 = [d] g [P“1(d)] < 0
and hence [P-1(d)] = —1 = [d]. Using Equation (2.5), we obtain that [P(P-1(d))] =
[P(d)], and hence [P(d)] = [d] = —1. Thus,

I‘(—1) = —1.
Now let n E Z " be given. Then

P(n) = P([d_"]), where —n E Z+

= P(Td]+[d]w:~-+[d1)

 

 

—n times
= poet) + r<1o1+--- + men
—n times
= —1-1—.. ~1=—n (—1)
—n times

Finally, let n E Z+ be given; then —n E Z”. Since I‘(-n) + P(n) = F(—n + n) =

P(n) = —F(—n) = —(—n) = n.
Therefore,

P(n) = n for all n E Z; and hence I‘ = 1.

Corollary 2.6 Let P be an order preserving field automorphism on L. Then P(r) =

r + 22:1 rkd" for all r E R, where rk E R for all k E Z+.

‘
...
‘ u
In...~
.
Al
r t
... ‘

 

T<

30

Proof. Let r > 1 be given in R. Then 1 = P(l) < P(r). If P(r) = r, we are
done. Assume P(r) 74 r; then P(r) — r 75 0. Since [P(r)] = [r] = 0, we have
that [P(r) — r] g 0. Assume [P(r) — r] = 0; then [P(r) — r] = [1], and hence
[P'1(P(r) — r)] = [P‘1(1)] = [1]. Thus [P'1(r) — r] = 0. If P(r) < r, then
r < P"1(r); so we may assume without loss of generality that r < P(r). Therefore

1 < r < P(r). Since [P(r)] = 0, there exists N E Z+ such that P(r) < N. Hence
1<r<P(r)<N<N-r<N-P(r). (2.8)
Since r < P(r) and [P(r) -— r] = 0, we have that
P(r)—r=t~r+irkdk, witht> 0 in R.

k=1

Let 6 be a rational number satisfying 0 < e < t. There exists k E Z+ such that

N < k - 6. Thus 6 - r < P(r) — r, and hence (1+ 6) - r < P(r). It follows that
(1+ 6)2 - r < (1+ 6) - P(r) = P((1+ 6)) - P(r) = P((1+ e) - r) < P(P(r)) = P2(r).
Using induction, we obtain that
(1+ 6)” < Pm(r) for all m E Z+.
In particular,
(1+N)-r< (1+k-e)-r< (1+e)’°or< Pk(r),

from which we obtain that

N - r < Pk(r). (2.9)
By Equation (2.8), P(r) < N. Hence P2(r) < P(N) = N. Using induction, it follows
that P'"(r) < N for all m E Z+. In particular, Pk(r) < N < N - r, which contradicts

Equation (2.9). So if r > 1 and P(r) 75 r, then [P(r) — r] < 0. It follows that

P(T‘) = T + Z’I‘kdk.

k=l

fl
a: Mu ['
“..‘tk

 

31

Hence the result is true for all r > 1 in R.

Now let r E R be such that O < r < 1; then r‘1 > 1. Thus,
P(r‘1)= r“1 + 2:3kd'c = r"1- (1+ r - Z skdk) = r“1 - (1+ 3),
k=l k=1
where ls] is infinitely small. Thus (1 + s)"1 = (1 + s’) where ]s’] is also infinitely

small. It follows that
P(r) = (P (r’1))—1 = r - (1 + s)'1= r + r - s’,

which proves the result for 0 < r < 1.

Since P(O) = 0 and P(l) = 1, the result is true for all r 2 0 in R. To show it is

true for r < 0 in R, we make use of the fact that P(r) = —P(—r).

Example 2.1 Define P : L —e L as follows: for :13 E L, write :1: = 2,2,3: ad" and
set P(cc) = 21:21:, 2kakd", where k2, = —[:1:]. Then P is an order preserving field

automorphism on L.

After this study of the properties of skeleton groups and field automorphisms, we
will now move on to introduce the Levi-Civita field ’R; and we will prove more results

about order preserving field automorphisms on ’R. in Section 3.3.

s eh.—
T

5.1.“...-

‘r u
n \ P'
”MI E 4

0w.

i]? 13.

1|» ‘ A»
h: n ‘
114‘??-
«Adult
‘s
e: I:
l
]r .
J “ w
. "-

‘vfl

I.
.I

s‘ .«v
3.] 1

§

1 O

.1 ‘.
V

Chapter 3

The Non-Archimedean Field R

In this chapter, we review the algebraic structure, the order structure and the topolog-
ical structure of the non-Archimedean Levi-Civita field R, which are found in [3, 5, 7]
We also review the differential algebraic structure of the field, which is useful for the

concept of differentiability [42].

3. 1 Algebraic Structure
We begin the discussion by introducing a specific family of sets.

Definition 3.1 (The Family of Left-Finite Sets) A subset M of Q is called left-
finite if and only if for every number r E Q there are only finitely many elements of
M that are smaller than 7‘. The set of all left-finite subsets of Q will be denoted by

f.
The next lemma gives some insight into the structure of left-finite sets.

Lemma 3.1 Let M E .7: be given. If M at 0, the elements of M can be arranged
in ascending order, and there exists a minimum of M. If M is infinite, the resulting

strictly increasing sequence is divergent.

32

l l
O‘?: — p _-
.$...*..CL ;

\
e-t

'7! r -
V \
1. 1..., 1‘s.

14..

..."’-n
w .33.].

lemma 1

-
r:
)

-
s
-.-

33

Proof. A finite totally ordered set can always be arranged in ascending order; so we

may assume that M is infinite.

For each n E Z+, set M7, = {33 E M : :1: S n}. Then, for all n, Mn is finite by
the left-finiteness of M, and we have that M = UnMn. Hence, we first arrange the
finitely many elements of M0 in ascending order, append the finitely many elements

of M1 not in M0 in ascending order, and continue inductively.

If the resulting strictly increasing sequence were bounded, there would also be a
rational bound below which there would be infinitely many elements of M, contrary
to the assumption that M is left-finite. Therefore, we conclude that the sequence is

divergent.

Lemma 3.2 Let M ,N E .7. Then the following are true.

oXCM=>XEf.

e MUNEF.

e MONEf.

e M+N={x+y:xEMandyEN}EJ-', andforeveryxEM+N, there

are only finitely many pairs (a, b) E M x N such that :1: = a + b.

Proof. The first three statements follow directly from the definition. For the proof
of the fourth statement, let :1: M, :1: N denote the smallest elements in M and N respec-

tively; these exist by Lemma 3.1. Let r E Q be given. Set

M“={a:EM]a3<r-:1:N}, N“={:1:EN|:v<r—:1:M},

M°=M\M“, N°=N\N“.

~p.,

pte

...5

'.

inn.

-

.5

‘. L
n

I?»

34

Then we have that

M+N

(M" u M") + (N“ U N”)
= (M“ +N“) u (M° + N“) u (M“ + N°) 11 (M°+ N0)

= (M"+N“)U(M°+N)U (M+N°).

By definition of M" and N°, we have that (M0 + N) and (M + N 0) do not contain
any elements smaller than r. Thus all elements of M + N that are smaller than r
must actually be contained in M “ + N “. Since both M “ and N u are finite because
of the left-finiteness of M and N, we obtain that M u + N “ is also finite. Thus there

are only finitely many elements in M + N that are smaller than r.

Finally, let :1: E M + N be given. Set r = a: +1 and define M“, N" as in the
preceding paragraph. Then we have that :1: E (M0 + N) and .7: E (M + N °). Hence

all pairs (a, b) E M x N that satisfy a: = a + b lie in the finite set Mu x N“.

Having discussed the family of left-finite sets, we introduce the following set of

functions from Q into R.

Definition 3.2 (The Set R) We define

R={f:Q-—>R:{x|f(x);é0}eJ-”}.

Hence, the elements of R are those real-valued functions on Q that are nonzero only

on a left-finite set, that is, they have left—finite support.

Remark 3.1 Since the desired field R is to be non-Archimedean and have roots of
positive elements (see Chapter 1), we infer using Theorem 2.2 that Q is the minimal
domain of definition of the elements of R in Definition 3.2. This already tells us

something about the uniqueness of R; see Theorem 3.11.

35

In the following, we denote elements of R by :12, y, etc. and identify their values
at q E Q with brackets, like in a:[q]. This avoids confusion when we later consider
functions on R. Since the elements of R are functions with left-finite support, it is

convenient to use the properties of left-finite sets in Lemma 3.1 for their description.

Definition 3.3 (Notation for Elements of R) An elements: of R is uniquely char-
acterized by an ascending ( finite or infinite) sequence (qn) of support points and a

corresponding sequence (a:[q,.]) of function values. We refer to the pair of sequences

((qn), (:1:[q,,])) as the table of :1:.

Already at this point it is worth noting that for questions of implementation, it
is usually sufficient to store only the first few of the support points and remember

carefully up to what “depth” a given number in R is known.

For subsequent discussion, it is convenient to introduce the following terminology.

Definition 3.4 (supp, A, ~, z, =,) For :13,y E R, we define
supp(:1:) = {q E Q : :13[q] 74 0} and call it the support of :13.
A(:1:) = min(supp(a:)) for :1: 51$ 0 (which exists because of left-finiteness) and
/\(O) = +00.
Comparing two elements, we say
:1: ~ y if and only if /\(x) = My);
:1: z y if and only if A(:1:) = /\(y) and :13[)\(a:)] = y[)\(y)];

:13 =,. y if and only if :1:[q] = y[q] for all q S r.

At this point, these definitions may feel somewhat arbitrary; but after having

introduced the concept of ordering on R, we will see that A describes “orders of infinite

. ~.r ‘
l'ff' A
‘3, . ..Au
w ' 'f
e

u
.. trvr‘l
‘ h

.‘ nul- '

lemme

‘ff
3‘ l I

“0'1

return

i ‘

36

largeness or smallness”, the relation “z” corresponds to agreement up to infinitely
small relative error, while “~” corresponds to agreement of order of magnitude and

is thus the same as the ~ introduced in Section 2.1.

Lemma 3.3 The relations ~, z and z, are equivalence relations. They satisfy
mzy¢x~w

and

ifa>binQ,then:1:-—-ay:r=by.

Definition 3.5 (The Number d) We define the number d E R as follows.

fld={1 Uq=1.

0 else

Apparently, the number d admits an n—th root for all n E Z+, denoted by dl/n

and given by

n 11 n =l
dl/ [q]=[0 else? n '

Also d has a multiplicative inverse denoted by d“1 and given by

fhd={l qu—l.

0 else
As we shall see, d plays the role of an infinitesimal and thus satisfies what Rall
suspected about the number (0,1) in his arithmetic of differentiation [33].

We now define arithmetic on R.

Definition 3.6 (Addition and Multiplication on R) We define addition on R

componentwise:

@+vmn=dd+nyWaUqEQ.

37

Multiplication is defined as follows. For q E Q, we set

(it y)[ql = X with] - slot]-

(11:: (]y 6 Q1

93 + Qy = q
We remark that R is closed under addition since supp(:1: + y) C supp (:13) U
supp (y), so by Lemma 3.2, we have that supp(:1: + y) is left-finite. Lemma 3.2 also
shows that only finitely many terms contribute to the sum in the definition of the
product. Furthermore, the product defined above is itself an element of R since the
sets of support points satisfy supp(:1: . y) C supp (x) + supp (y); so that application

of Lemma 3.2 shows that supp(:1: - y) E f.

It turns out that the operations + and - we just defined on R make (R,+,-) into
a field (see Theorem 3.4 below).

Theorem 3.1 (R,+,-) is a commutative ring with a unit.

Proof. The proof is straightforward, and we leave it as an exercise for the reader to

fill in the details.

As it turns out, R can be viewed as an extension of R.

Theorem 3.2 (Embedding of R into R) R can be embedded into R under the

preservation of its arithmetic structure.

Proof. Let :1: E R. Define II : R —> R by

Hie)h]=[‘3 13:3.

Then II is one to one, and direct calculation shows that

II(:1: + y) = H(:1:) + H(y) and

“(x - y) = HOB) - II(y)-

38

So R is embedded as a subfield in the ring R. However, the embedding is not surjec-

tive, since only elements with support {0} are actually reached.

Remark 3.2 In the following, we identify an element :1: E R with its image II(:1:) E R

under the embedding.
We also make the following observation.

Remark 3.3 Let 2:1 and 3:2 be real numbers. Then if both x1 and 132 are nonzero, we

have that 2:1 ~ :132. Furthermore, :r1 4:: 3:2 is equivalent to 2:1 = 3:2.

The only nontrivial step toward the proof that R is a field is the existence of
multiplicative inverses of nonzero elements. For this purpose, we prove a new theorem

that will be of key importance for a variety of proofs and applications.

Theorem 3.3 (Fixed Point Theorem) Let qM E Q be given. Define M C R to
be the set of all elements :1: of R such that Mr) 2 qM. Let f : M —> R satisfy
f (M) C M. Suppose there exists It > 0 in Q such that for all $1,332 E M and for all

q E Q, we have that
$1 :9 552 => f($1)=q+k “932)-

Then there exists a unique solution :1: E M of the fixed point equation
x = f (:13)
Proof. We choose an arbitrary a0 E M and define recursively
a1, = f(an_1), for n = 1,2,

Since f maps M into itself, this generates a sequence of elements of M. First we

show that for all n E Z +, we have that

an[p] = an_1[p] for all p < (n —- 1)k + qM. ' (3.1)

39

Since ao,a1 E M, we have that a1[p] = O = ao[p] for all p < qM. So Equation (3.1)
holds for n = 1. Assume it is true for n = m; we show it is true for n = m + 1. Thus,
we have that

am[p] = am_1]p] for all p < (m — 1)]: + qM. (3.2)
Let t < mk + qM be given. Then t -— k < (m — 1)k + qM; and hence Equation (3.2)

entails that

am =t—lc am—l -

Hence

am+l = f(am) =t f(am—l) = arm

which entails that

am+1[p] = am[p] for all p S t.

This is true for all t < mk + qM; hence

amelb] = am[p] for all p < mk + qM.

Thus, Equation (3.1) is true for n = m + 1, and hence it is true for all n E Z+.

Next we define a function a: : Q —+ R in the following way. For q E Q choose
11 2 1 such that (n — 1)k+ qM > q. Set :1:[q] := an [q]; note that, by virtue of Equation
(3.1), this is independent of the choice of n. Phrthermore, we have that a: =q an.
So in particular :1: is an element of R since for every q E Q, the set of its support
points smaller than q agrees with the set of support points smaller than q of one of

the an E M. Also, since :1:[p] = 0 for all p < qM, we obtain that :13 E M.

Now we show that :1: defined as above is a solution of the fixed point equation.
For q E Q choose again n 2 1 such that (n — 1)lc + qM > q. Then it follows that :c =q

an =q an“. By the contraction property of f, we thus obtain that f (:13) =q+k f (an),

40

which in turn implies that

$in = an+1lql = f (an)[q} = f (3:)qu-
Since this holds for all q E Q, we have that :13 is a fixed point of f.

It remains to show that :r is a unique fixed point of f in M. Assume that y E M

is a fixed point of f. The contraction property of f is equivalent to
A(f($1) — f(Tg» Z /\(£L'1- $2) + k for all 2131,1132 6 M.
This implies that

/\($ - y) = A(f(x) - ND) 2 Ma: - y) + k.

which is possible only if y = 11:, since k > 0.

Remark 3.4 Without further knowledge about R, the requirements and meaning of
the fixed point theorem are not very intuitive. However, as we will see later, the
assumption about f means that f is a contracting function with an infinitely small
contraction factor. Furthermore, the sequence (an) that is constructed in the proof is
indeed a Cauchy sequence, which is assured convergence because of the Cauchy com-
pleteness of R with respect to its order topology, as discussed in Chapter 4. However,
while making the situation more transparent, the properties of ordering and Cauchy
completeness are not required to formulate and prove the fixed point theorem, and so

we refrained from invoking them here.

It is also worthwhile to point out that, in spite of the iterative character of the fixed
point theorem, for every q E Q the value of the fixed point :1: at q can be calculated in

finitely many steps. This is of significant importance especially for practical purposes.

Using the fixed point theorem, we can now easily show the existence of multiplica-

tive inverses.

41

Theorem 3.4 (R, +,- ) is a field.

Proof. It remains to show the existence of multiplicative inverses of nonzero elements.
So let :1: E R \ {0} be given. Set q = Mac), a = :1:[q] and :1." = l/a - d‘q - :13. Then
Mx‘) = O and 111:“ [O] = 1. If an inverse of :1:" exists, then l/a - d‘q - (:1:")"1 is an inverse

of :13; so without loss of generality, we may assume that Mr) = O and :r[0] = 1.

If :1: = 1, there exists an inverse. Otherwise, :1: is of the form :1: = 1 + y with
0 < k = My) < +00. It suffices to find 2 E R such that (1+ 2) - (1+ y) = 1. This is

equivalent to

z = —y - z — y.
Setting f (z) = -y - z — y reduces the problem to finding a fixed point of f. Let
M = {z E R: Mz) _>_ k}. Let z E M be given; then

My - z) > My); and hence Mf(z)) = My) = k.

Hence f(M) C M. Now let z1,22 E M satisfying 21 =q 22 be given. Since My) = k,

we obtain that y - zl =q+k y - 22, and hence
-y-zi-y=e+t —y-z2—y-
Thus f satisfies the hypothesis of the fixed point theorem (Theorem 3.3), and conse-

quently a fixed point of f exists. This finishes the proof of Theorem 3.4.

Now we examine the existence of roots in R. Using the fixed point theorem, we

show that, regarding this important property, the new field behaves just like R.

Theorem 3.5 Let a: E R be nonzero, and set q = Mrs). If n is even and :13[q] is
positive, then :1: has two nth roots in R. If n is even and :1:[q] is negative, then :1: has

no nth roots in R. If n is odd, then :1: has a unique nth root in R.

42

Proof. Let :1: be a nonzero number and write x = a - d9 - (1+ y), where a E R, q E Q,
and My) > 0. Assume that w is an nth root of :1:. Since q = Mz) = Mw") = nMw),
we can write w = b - d9/" - (1 + z), where b E R and Mz) > 0. Raising to the nth
power, we see that b" = a and (1+ 2)" = 1 + y have to hold simultaneously. The first
of these equations has a solution if and only if the corresponding roots exist in R. So

it suffices to show that the equation
(1+ z)" 2 1+ y (3.3)

has a unique solution with Mz) > 0. But this equation is equivalent to nz+z2-P(z) =
y, where P(z) is a polynomial with integer coefiicients. The equation can be rewritten

as a fixed point problem 2 = f (z), where

 

Let

M={zER:Mz)ZMy)}.

For all z E M, we have that Mz) Z My) > 0. Thus, MP(z)) Z 0, and hence

M22 ' P(l)) = 2M2) + MP(Z» > M?!) Z My)-
Hence we obtain that f(z) % y/n; so f(z) E M. Hence
f(M) C M.

Now let 21,22 E M satisfying 21 :9 22 be given. Then M21) 2 My), Mz2) _>_
My), and the definition of multiplication shows that we obtain 2? =q+,\(y) 2%. By
induction on m, we obtain that 21" =q+,\(y) 2;" for all m > 1. In particular, this
implies zf -P(z1) =q+,\(y) 2% - P(Zg) and finally f(zl) =q+,\(y) f(Zg). So f and M
satisfy the hypothesis of the fixed point theorem which provides a unique solution of

(1+z)" = 1 +y in M and hence in R.

43

We remark that a crucial point to the proof was the existence of roots of the
numbers (1"; hence we could not have chosen anything smaller than Q as the domain

of the functions that are the elements of our new field.

We end this section by remarking that the field C, obtained by adjoining the
imaginary number i to R, is algebraically closed. Although a rather deep result, it
is obtained with limited effort using the fixed point theorem as well as the algebraic

completeness of C' (see [5]).

3.2 Differential Algebraic Structure

We introduce an operator 8 on R and show that it is a derivation.

Definition 3.7 Define 6 : R —> R by

(fl-rho] = (q + 1)qu + 1].

Lemma 3.4 (9 is a derivation on R; that is
6(x+y) = 82:+8y and 8(x-y) = (023)-y+2:-(0y) for allx,y E R.
Thus, (R, +, -,6) is a diflerential algebraic field. Furthermore, we have that

Mars) = M2:) — lif Mr) 7é 0,00 and

60 = 0;
but if M2:) = 0, then M623) can be either greater than, equal to, or smaller than Mr).

Proof. Let 2:, y E R and let q E Q be given. Then

(8(93 + 11)) [q] = (q +1)(-’r + y)[q +1] = (q +1)qu + 1] + (q +1)qu +1]

= (51:)[ql + (Be/Mol-

44

This is true for all q E Q; hence 0(2: + y) = 82: + 8y.

For all q E Q, we also have that

(5(05 - 31)) [q] = (q +1)(x-y)[q +1]

= (q + 1) Z while/[<12]

or + <12 = q + 1
91 6 Iupp(z).q'2 E -upp(y)

= 2: (q + 1)$[qily[q2l

<11 + 92 = <1 + 1
<11 6 nIr>1=(=).<12 6 '“PP(U)

= Z (Q1$[91ly[Q2l + $[q1]Q2$[Q2])

q1 + <12 = e + 1
or 6 -ur>p(z).<12 E Iuppm

= Z q12:[q1]y[q2] + Z 17 [Q1l(12y[€12l

91+92=q+1 91+92=9+1
¢11 E IuPP(=).¢n 6 lmwho) tn 6 mpp(z). <12 6 aupp(v)
= Z (s +1):1:[s +1]y]t] +

a + t = 11
s + 1 E uupp(==). t E IUPPUI)

Z 2:[s](t + 1)y[t + 1]

s + t = q
s E supp“). t + l E IUPPW)

= Z (82:)[s]y[t] + Z 2:18] (6.211111

5 + t = q a + t = q
a 6 mppwz). t 6 wppw) 4- E wpph). t E suppwv)

= ((01?) ~11) [q] + (1‘- (3.11)) [q]
== ((5%) - y + a: - (011)) lol-
This is true for all q E Q; and hence 6(2: - y) = (02:) - y + 2: - (6y).

Now let 2: E R be given such that M23) 79 0,00. Then for all g < M23) — 1, we

have that q + 1 < Mr); and hence
(51:)[ql = (q + 1)qu + 1] = 0-
Hence M62) 2 Mr) - 1; but

(averse) — 11 = Are-Win] 7e 0.

45
Hence, M1923) = M2:) — 1.
On the other hand, we have that
(80)]q] = (q + 1)O[q +1] = 0 for all q E Q.

Thus,
00 = O; and hence M00) = M0) = 00.

To prove the last statement, let
2:1: 1, 2:2 =1+d, and 2:3 = 1 +d1/2; then M2:,-) = 0 for j = 1,2,3.
We have that

62:1 = 0, and hence M621) > M231);
82:2 = 1, and hence deg) 2 M22);

82:3 2 éd’lfl, and hence M623) < M23).

3.3 Order Structure

In the previous section we showed that R does not differ significantly from R as far

as its algebraic properties are concerned. In this section we discuss the ordering.

The simplest way of introducing an order is to define a set of “positive” numbers.

Definition 3.8 (The Set R+) Let R+ be the set of all nonvanishing elements 23 of

R that satisfy 2:]M2:)] > 0.

Lemma 3.5 (Properties of R+) The set R+ has the following properties.
72+ r1(—R+) = (1, 72+ r1{0} = 0, and 72+ U {0} u (—R+) = R,-

:1:,yER+ => 113+yER+ andx-yER+.

46

The proofs follow rather directly from the respective definitions.

Having defined R+, we can now easily introduce an order in R.

Definition 3.9 (Ordering in R) Let 2:,y E R be distinct. We say :1: > y if and

only ifa: - y E R+. Furthermore, we say :1: < y if and only ify > 2:.

With this definition of the order relation, R is a totally ordered field.

Theorem 3.6 (Properties of the Order) With the order relation defined in Def-

inition 3.9, (R,+,-) becomes a totally ordered field.

Furthermore, the order is compatible with the algebraic structure of R, that is, for

any :13, y, z E R, we have that

2:>y => 2:+z>y+z; and

2:>y => :1:~z>y-zifz>0.

Since the proof follows the same arguments as the corresponding ones for R, the
details are omitted here. We immediately obtain that the embedding H in Theorem

3.2 is compatible with the ordering, that is
2: < y => II(2:) < H(y).

Furthermore C, like C, cannot be ordered.

Thus R, like C, is a proper field extension of R. Note that this is not a contra—
diction of the well-known uniqueness of C as a field extension of R. The respective
theorem of Frobenius asserts only the nonexistence of any (commutative) field on R"

for n > 2. However, regarded as an R-vector space, R is infinite dimensional.

Besides the usual order relations, some other notations are convenient.

47

Definition 3.10 (<<, >>) Let a,b E R be nonnegative. We say that a is infinitely
smaller than b (and write a << b) if and only ifn - a < b for all n E Z+; we say
that a is infinitely larger than b (and write a >> b) if and only if b < a. If a < 1,
we say that a is infinitely small; if 1 < a, we say that a is infinitely large. Infinitely
small numbers are also called infinitesimals or differentials. Infinitely large numbers
are also called infinite. Nonnegative numbers that are neither infinitely small nor

infinitely large are also called finite.

Corollary 3.1 For all a, b,c E R+, we have that

a<<b => a<b, and

a<<bandb<<c : a<<c.
Moreover, we observe that

61" << 1 if and only ifq > 0, and 11" >1 if and only ifq < 0.
Corollary 3.2 The field R is non-Archimedean.

Proof. We have that n - d < 1 for all n E 2+; and hence d 74 1.

By Lemma 2.2, every field automorphism on R is order preserving. The following
example shows that, while the identity map is the only field automorphism on R
by Corollary 2.2, there are nontrivial field automorphisms on R. However, Theorem
3.7 below shows that the image of a real number under a field automorphism is

approximately equal to the number itself.

Example 3.1 Define P : R —e R as follows: For :1: E R, write :1: = 20631117110)“qu

and set P(m) = qusupp(z) aqd3". Then P is a field automorphism on R.

48

Remark 3.5 Note that, in Example 3.1, P(d) = d3 76 d, in contrast with Corollary
2.5 and Example 2.1.

Theorem 3.7 Let P be a field automorphism on R. Then P(r) z r for all r E R.

Proof. By Lemma 2.2, we have that P is order preserving. Then the proof is exactly

the same as that of Corollary 2.6.

It is a crucial property of the field R that the differentials, especially the formerly
defined number 11, satisfy Leibniz’s intuitive idea of derivatives as differential quo-
tients. This will be discussed in great detail in Chapter 5; but already here we want

to give a simple example.

Example 3.2 (Calculation of Derivatives with Differentials) Let f : R —> R

be given by f (x) = :1:2 — 2x.

Obviously, f is differentiable on R, and we have that f’ (x) = 2x — 2 for all x E R.
As we know, we can obtain certain approximations to the derivative at the position

x by calculating the difference quotient

f(rr + Aiv) - f(x)
Ax

 

at x. Roughly speaking, the accuracy increases if Ax gets smaller. In our enlarged
field R, infinitely small quantities are available, and thus it is natural to calculate the
difference quotient for such infinitely small numbers. For example, if we let Ax = d

and let f denote the continuation of f to R, then we obtain that

f(x+dc)l—f(x) ___ (x2+2xd+d2-2:—2d)-(x2—2x) =2x—2+d.

 

 

We realize that the difference quotient differs from the exact value of the derivative

by only an infinitely small error. If all we are interested in is the usual real derivative

49

of the real function f, then this is given exactly by the “real part” of the difference

quotient.

3.4 Topological Structure

In this section we examine the topological structures of R and the related sets. We
will see that on R, in contrast to R, several different nontrivial topologies can be

defined, all of which have certain advantages.

We begin with the introduction of an absolute value; this is done as in any totally

ordered field.

Definition 3.11 (Absolute Value on R) Let x E R be given. We define the ab-

solute value of x as follows.

x ix>0
I,I={ 1 __

—x ifx<0°

Lemma 3.6 (Properties of the Absolute Value) The mapping | | : R —e R has

the following properties.

]x] = 0 if and only ifx = 0.
ltv - HI = let] - To! for all 2:,y 6 R-
]x + y] S ]x] + ]y] for all 2:,y E R.

lel - lyll S Inc - v] for all 16.11 6 R-

Proof. The proof follows the same lines as the proof of the corresponding result in

R.

Just as in any totally ordered set, we can now introduce the so—called order topol-

ogy.

50

Definition 3.12 (Order Topology) We call a subset M of R open if and only if
for any x0 E M there exists an e > 0 in R such that 0(x0, e), the set of points x with

Ix — x0] < e, is a subset of M.
Thus all e—balls form a basis of the topology. We obtain the following theorem.

Theorem 3.8 (Properties of the Order Topology) With the above topology, R
is a nonconnected topological space. It is Hausdorff. There are no countable bases.

The topology induced to R is the discrete topology. The topology is not locally compact.

Proof. We first observe that for all 220 E R and for all e > 0 in R, the balls 0(x0, e)
are open; and so is the whole space. Birthermore, all unions and finite intersections

of open sets are obviously open. To show that R is not connected, let

M1 = {xEszSOor(x>0andx<<1)}; and

M2 = {xER:x>0andx5K1}.

Then M1 and M2 are open and disjoint; moreover, we have that M1 LJ M2 = R.

For all 2:,y E R, 0(x, ]x -— yI/2) and 0(y, |x — yI/2) are open and disjoint, and

they contain 2: and y, respectively. Hence R is Hausdorff.

There can not be any countable bases because the uncountably many open sets
M x = 0(X,d), with X E R, are disjoint. The open sets induced on R by the sets
M X are just the single points. Thus, in the induced topology, all sets are open and

the topology is therefore discrete.

To prove that the space is not locally compact, let x E R be given and let U be
a neighborhood of x. We show that the closure I7 of U is not compact. Let 6 > 0 in

R be such that 0(x, e) C U and consider the sets

M_1={yER:y<xory—x>>d-e};

51
M,, = (x+(n-—1)d-e,x+(n+1)d-e) forn=0,1,2,....
Then it is easy to check that Mn is open for all n 2 —1 and
Uf=_1M,, = R; in particular, U C Uf,°=_1M,,.

But it is impossible to select finitely many of the Ma’s to cover (7 because each of the
infinitely many elements x + nd - e of U, n = —1,0, 1, 2, . . ., is contained only in the

set Mn.

Remark 3.6 A detailed study of the properties in Theorem 3.8 reveals that they hold
in an identical way on any other non-Archimedean structure, and thus the above

unusual properties are not specific to R.

Besides the absolute value, it is useful to introduce a semi-norm that is not based
on the order. For this purpose, we regard R as a space of functions as in the beginning,

and define the semi-norm as a mapping from R into R.

Definition 3.13 Given r E Q, we define a mapping || - ]|,. : R —> R as follows.
llmllr == SUPflIUTQTl I <1 S 1'l- (34)

Remark 3.7 The supremum in Equation (3.4) is finite and it is even a maximum

since, for any r, only finitely many of the x[q] ’3 considered do not vanish.

Lemma 3.7 For any r E Q, the mapping I] ~ I], : R —> R satisfies the following.

]|0||r = 0 = ][d‘]],. for allt > r in Q. (3.5)
“x“, = I] — x“, for all x E R. (3.6)
”x”, Z 0 for all x E R. (3.7)
H33 + yllr S ll-Tilr + llyllr f0r all 1‘, y E 73- (3-8)

lllxllr - Ilyllrl S llx - yllr for all any 6 73- (3-9)

Li A ‘
Ll 1.?r4

.. k“

-..' ,.
."$¢‘

.LV‘: Fin-n
8.1; r *
‘ .

52

Proof. Equations (3.5), (3.6), and (3.7) follow readily from the definition. To prove
Equation (3.8), let x, y E R and r E Q be given. Then

llxllr = SHPH-TTQHIQST},

sun{|y[q]| = q S 7‘}. and

llyllr

ll$+yllr = sup{l(:v+y)lql|=qu}-

Let go E Q be such that go 3 r and ](x + y)[qo]| = “x + 31“,. Then

“5'7 + yllr = [(33 + y)l€10l] = lxlqo] + yiQOll
S [Thrall + lquOll

S llffill.~ + llyllr.

We finally prove Equation (3.9): Let x, y E R and r E Q be given. Then, using

Equation (3.8), we have that ”x“, g “x — y”, + Ilyllr, from which we obtain that
little — llyllr S III? - yllr. (3-10)
Interchanging x and y in Equation (3.10) and using Equation (3.6), we obtain that
llyllr -||1‘||r S Hy - (Elle = “93 - yllr- (3-11)

combining Equation (3.10) and Equation (3.11), we obtain Equation (3.9) and finish

the proof of the lemma.

Remark 3.8 Fl‘om Equations (3.5), (3.6), (3.7), and {3.8), we infer that I] - II, is a

semi-norm but not a norm, for any r E Q.

The topology induced by the family of these semi-norms will be called weak topol-

ogy.

53

Definition 3.14 (Weak Topology) We call a subset M of R open with respect to
the weak topology if and only if for any x0 E M there exists 6 > 0 in R such that
S(xo,6) = {x E R: IIx — $0II1/e < e} C M.

We will see that the weak topology is the most useful topology for considering
convergence in general; see Chapters 4, 5, 6, and 7. Moreover, it is of great importance

for the implementation of the R calculus on computers; see Chapter 7.

Theorem 3.9 (Properties of the Weak Topology) With the above definition of
the weak topology, R is a topological space. It is Hausdorff with countable bases. The

topology induced on R by the weak topology is the usual order topology on R.

Proof. It is easy to check that for all 230 E R and for all e > 0 in R, the balls S (x0, 6)
are open; and so is the whole space. Phrthermore, all unions and finite intersections
of open sets are open. The halls S (r, q) with r, q E Q form a countable basis of the
topology. We obtain a Hausdorff space: Let x, y E R be given, let r = Mx — y), and

let

 

6 = min { |(x -2y)[7"l|, 2'1“} .

Then S (x, e) and S (y, e) are disjoint and open, and they contain x and y, respectively.
Finally, considering elements of R, their supports are all equal to {0}. Therefore, the
open subsets of R in the weak topology correspond to the open subsets of R in its

order topology.

In Chapter 4, we will study in details convergence of sequences and series in the
strong and weak topologies, and we will show that R is Cauchy complete with respect

to the strong topology while it is not with respect to the weak topology.

In addition to the two topologies discussed above, there is another topology which

takes into account that, in any practical scenario, it will not be possible to detect

 

54

infinitely small errors, nor will it possible to measure infinitely large quantities. We

obtain this topology by a suitable continuation of the order topology on R.

Definition 3.15 (Measure Topology) Given any open subset of R, we form a sub-
set of R containing the elements of the original set as well as all the elements infinitely
close to them. To the family of sets obtained this way, we add one more set, namely

the one containing every element with infinitely large absolute value.

Thus a basis of this topology consists of all e—balls with real 6 and the set of

numbers with infinitely large absolute value.

Theorem 3.10 (Properties of the Measure Topology) With the above topology,
R is a nonconnected topological space with countable bases. It is not Hausdorfi'. The

topology is locally compact and induces the usual order topology on R.

Proof. We can directly show that the whole space as well as unions and finite
intersections of open sets are open. Obviously, elements with infinitely small difference
can not be separated; they are always simultaneously inside or outside of any given
open set. Hence the space is not Hausdorff with respect to the measure topology.

The rest follows by transferring the properties of the order topology on R.

Remark 3.9 (Comparison of the Topologies) The order topology is a refinement

of both the weak topology and the measure topology.

To finish this section, we will show that the field R is indeed the smallest non-
Archimedean extension of R satisfying the basic requirements demanded in Chapter

1, which gives it a unique position among all other field extensions.

E.

 

55

Theorem 3.11 (Uniqueness of R) The field R is the smallest totally ordered non-
Archimedean field extension of R that is complete with respect to the order topology, in
which every positive number has an nth root, and in which there is a positive infinitely

small element a such that (an) is a null sequence with respect to the order topology.

Proof. Obviously, R satisfies the conditions above. So it remains to show that R
can be embedded in any other field extension of R that has the properties mentioned '9:

above. So let S be such a field.

Let 6 E S be positive and infinitely small such that (6") is a null sequence with

‘A‘.’ .

respect to the order topology in 8. Let 61/" be an n—th root of 6. Such a root exists

 

tar.-

according to the requirements. Now observe that
(61/71)," = (61/(np))mp’ for all p E z+.
So let q = m/n be given in Q, and let
6" = (51/")"’.
This element is unique. Furthermore, 6‘I is infinitely small for q > 0. Let ql < (12 be
given in Q. Then we clearly have that 6"1 > 692. Now let a E R be given. Then we
also have that a E S, and hence a - 6" E 8.

Now let ((q,-), x[q,-]) be the table of an element x of R. Consider the sequence

1'

s,- = Z qu,]6q".

i=1

Then the sequence (Sj) converges in 8: Let c > 0 be given in 8. Since the sequence

(6”) is a null sequence in 8, there exists N E Z + such that
[6"] < e for all V 2 N.
Since the sequence (q,) is strictly divergent, there exists N1 E Z + such that

q,- 2 Nfor alliZ N1.

56

But then we have for arbitrary jl > jg 2 N1 that

 

i=J'2 i=J'2 i=j2

jl j1 jl
[311—312] = 1293045“ 5 Z IxTCh-llé‘“ 5 (Z: lxlqtlll 6%“

.71
g (E ]x[q,-]I) 6N+1 < 6” < e.

i=j2
Thus the sequence (3,) is Cauchy, and hence it converges in S because of the Cauchy
completeness of S. We now assign to every element 23;, qu,-] - d“ of R the element
2,921 qui] - 64" of S. The mapping is one to one. Furthermore, we can easily verify

that it is compatible with the algebraic operations and the order on R.

Remark 3.10 A field with the properties of R could also be obtained by successively
extending a simpler non-Archimedean field, e. g. the well-known field of rational func-
tions. To do this, we first would have to Cauchy complete the field. After that, the
algebraic closure has to be done, for example by the method of Kronecker-Steinitz.
This method, however, is nonconstructive, whereas the direct path followed here is

entirely constructive.

Remark 3.11 In the proof of the uniqueness, we noted that 6 was only required to
be positive, infinitely small and such that (6") is a null sequence in 8. But besides
that, its actual magnitude was irrelevant. Thus, none of the infinitely small quantities
is significantly different from the others. In particular, there exists a nontrivial field
automorphism on S. This remarkable property has no analogy in R where the identity

map is the only field automorphism; see Corollary 2.2.

--.. , ”l

 

Chapter 4

Sequences and Series re

In this chapter, we review convergence of sequences and series with respect to the

r—- “I!" n1.“
1

order and weak topologies following [3, 5, 7]. We also prove new results; in particular,
those dealing with the convergence of sums and products of sequences and series. We
then enhance and prove a weak convergence criterion for power series, Theorem 4.12,
and use that to extend the transcendental functions to R and study their properties

in Section 4.4.

4.1 Strong Convergence
We begin this section by studying a special property of sequences.

Definition 4.1 (Regularity) A sequence (3") in R is called regular if and only if
the union of the supports of all members of the sequence is a left-finite set, that is if

and only if U°° supp(s,,) E .7.

n=0

This property is not automatically assured, as becomes apparent from consider-
ing the sequence (d‘”). As the next theorem shows, the property of regularity is

compatible with the common operations of sequences.

57

Lemma 4.1
11. Then the
:5 rd! as an;

"35,: 3 tn ar-

Proof. Let

requirements .

Every 511;:
Lillie 8,, 01' (j
t“0311,!ther

The Stippt-I

\

 

 

eons of the i

Dtfinlllon 4

5 - " 5‘ I
'3: L5 :tTOngi

3* (l' ’
"Efir-E ‘ C . *
\

[Q r,
eugg ti“)
lift e1
. » “at i8
k.
lift-S Q:

58

Lemma 4.1 (Properties of Regularity) Let (3,.) and (tn) be regular sequences in
R. Then the sequence of the sums, the sequence of the products, any rearrangement,
as well as any subsequence of one of the sequences, and the merged sequence r2,, = sn,

r2n+1 = t1, are regular.

Proof. Let A = Ufiiosupp(sn) and B = Uziosuppan). Then, according to the

requirements, we have that A E f and B E 1:.

Every support point of the sequence of the sums is a support point of either one
of the sn or one of the tn and is thus contained in (A U B) E 1:, using Lemma 3.2.
Every support point of the sequence of the products is contained in (A + B) E 7:,

again using Lemma 3.2.

The support points of any subsequence of (3") are contained in A , and the support

points of the joined sequence (rn) are contained in A U B.

Definition 4.2 (Strong Convergence) Let (5") be a sequence in R. We say that
(3”) is strongly convergent to the limit 3 E R if and only if for every 6 > 0 in R there
exists N E Z+ such that

]sn—s]<eforalanN.

Remark 4.1 Like in any other metric space, it is easy to show that if the limit of a

sequence exists then it is unique.

Using the idea of strong convergence allows a simple representation of the elements
of R that is indeed strongly reminiscent of the familiar expansion of real numbers in

powers of ten, and that enjoys a similar usefulness for practical calculations.

59

Theorem 4.1 (Expansion in Powers of Differentials) Let ((qn), (qu..])) be the
table of x E R (see Definition 3.3). Then the sequence (23"), given by xn = 2;, x[q,-]-
dq‘ for all n E Z+, converges strongly to the limit x. Hence we can write

00

x = Zqun] edq".

1121

Proof. Without loss of generality, we may assume that the set {qn} is infinite. Let
6 > O in R be given. Choose N1 E Z + such that le < 6. Since (qn) diverges strictly

according to Lemma 3.1, there exists N E Z + such that
qn > N1 for all n _>_ N.
Hence we have that
(x,, — x)[t] = O for all t 5 N1 and for all n 2 N.

Thus

Ixn—x]<eforalanN.

Therefore, (xn) converges strongly to x.

Lemma 4.2 Let (3“) be a sequence converging strongly in R to 3. Then (IsnI) con-

verges strongly in R to Is]

Proof. Let 6 > 0 in R be given. Then there exists N E Z+ such that Ism — s < e

for all m 2 N. Therefore,
IIsmI — Is]] 3 Is,,, — s] < e for all m 2 N.

Hence, (IsnI) converges strongly to Is].

Remark 4.2 The converse of Lemma 4.2 is not true.

60

Proof. Consider the sequence (3,.) in R, where, for k = 0,1,..., 32,, = —1 and
32;,“ = 1. Then Isn] = 1 for all n 2 0. Hence (IsnI) converges strongly to 1 in R.

However, (3") does not converge strongly in R.

Definition 4.3 Let (3,.) be a sequence in R. Then we say that (3") is strongly Cauchy

if and only if for all e > 0 in R, there exists N E Z+ such that

Ism — 3;] < e for all m,l Z N.

Like R, the new field R is Cauchy complete with respect to the order topology.

That is, a sequence (sn) in R converges strongly if and only if (3") is strongly Cauchy.
Theorem 4.2 R is Cauchy complete with respect to the order topology.

Proof. Let (3") be a sequence in R that converges strongly to s E R. We show that
(3“) is strongly Cauchy. So let 6 > 0 be given in R. Then there exists N E Z + such
that

Isn—sI<-;-foralln2N.

For all m,l Z N, we have that

e e
Ism—sll=]sm—s—(s;—s)lgIsm—sI+Is;—sI<-§+§=e.

Hence (3") is strongly Cauchy.
Now let (3,.) be a strongly Cauchy sequence in R. We show that (sn) converges

strongly in R. For all r E Q, there exists N, E Z + such that

Is", — 3;] < dT+l for all m,l 2 N,.. (4.1)

Thus.

B1; Equation

leheszQ

 

apple} is let]

11.3]. we 0h?

It‘s there a

 

1"; r! .
l’ht‘tmte. Fl

Li 7i. There 1

hence

61

Thus,
smIq] = 3N, [q] for all m 2 N, and for all q s r. (4.2)
By Equation (4.1), we may assume that

er S er If 7‘1 < T2. (4.3)

Define s : Q —+ R by qu] = qu [q] First we show that s E R; that is, we show that
supp(s) is left-finite. So let r E Q be given. Combining Equation (4.2) and Equation

(4.3), we obtain that

quI = SN, [(1] = SN. [(1] for all q S r.

Thus there are only finitely many q S r such that qu] 31$ 0, and hence supp(s) is
left-finite. Finally, we show that (s,.) converges strongly to 3. So let 6 > 0 be given

in R. There exists r E Q such that d' < 6. Then
smIq] = s N, [q] = s[q] for all m 2 N, and for all q S r.

Hence

Ism—sI<<d’<eforallmZN,.
Thus, (3,.) converges strongly to s in R.

Like in any other metric space, every Cauchy sequence is bounded.
Lemma 4.3 Let (s,.) be a strongly Cauchy sequence in R. Then (3,.) is bounded.
Corollary 4.1 Every strongly convergent sequence in R is bounded.

Theorem 4.3 (Strong Convergence Criterion for sequences) Let (s,.) be a se-
quence in R. Then (3,.) converges strongly in R if and only if for all r E Q there
exists N E Z+ such that

s... =, s; for all m,l Z N.

Proof. Let

Then there r

Tzen there e

nn3~f

 

limo '
.8 (so) it

rm»
e-‘fl't‘E‘a .
":95 Sir]

Lemma 4.4

_e’.’

62

Proof. Let (s,.) be a strongly convergent sequence in R, and let r E Q be given.

Then there exists N E Z + such that
Is... — 3,] < d”1 for all m,l Z N.

Hence 3... =,. s. for all m,l Z N.

Now let (3,.) be a sequence in R such that for all r E Q there exists N E Z + such

that s,,. =, s; for all m,l Z N. Let 6 > 0 in R be given; and let
r = Me).
Then there exists N E 2‘“ such that s,,. =,. s, for all m,l 2 N; and hence
|s,,. — 3,] << d' for all m,l 2 N.
Since d" ~ 6, we obtain that
|s,,. — 81] << 6 for all m,l 2 N.

Hence (3,.) is a strongly Cauchy sequence; so by Theorem 4.2, the sequence (3,.)

converges strongly in R.
Lemma 4.4 Let (s,.) be a strongly convergent sequence in R. Then (3,.) is regular.

Proof. Let s be the limit of (s,.) in R; and let r E Q be given. Then there exists

N E Z+ such that

Ism-sI <d'+1 for allmZN.

It follows that

s,,.[q] = s[q] for all m 2 N and for all q S r.

Thus,

(-00,r] fl (Uf=osupp(s,.)) = (—00,r] fl (Uggolsupp(s,.)) U ((-00,r] fl supp(s))

finite. it b4
the sequence

The foils
to not hold ;

Theorem 4

Proof. Let I

ecst N e Z

(I)

p

flflmz‘

NOW agx";

Then there r

15%.!) \'

l
we have they

 

 

 

63
is finite, it being the union of two finite sets. So Uffzosupp(s,.) is left-finite, and hence
the sequence (3,.) is regular.

The following results: Theorem 4.4, Corollary 4.2, Corollary 4.3 and Corollary 4.4

do not hold in R; the non-Archimedicity of R is the key to their proofs.

Theorem 4.4 Let (s,.) be a sequence in R. Then (3,.) is strongly Cauchy if and only

if (3.....1 — 3,.) is a null sequence.

”flat-n I
pl

Proof. Let (s,.) be a Cauchy sequence in R, and let 6 > 0 in R be given. Then there

exists N E Z+ such that Is; — 3...] < e for all l,m 2 N. In particular, Ism+1 — 3",] < e

 

for all m 2 N. Hence, limnxoo(s,.+1 — 3,.) = 0.
Now assume that (s,.+1 — s,.) is a null sequence in R, and let 6 > 0 in R be given.
Then there exists N E Z + such that

Ism+1 — 3...] < de for all m 2 N.

Let k,l Z N be given. Without loss of generality, we may assume that k > l. Then

we have that

Ist-Szl = ISt—sh_i+sh_1—8e_2+---+31+i-81|
S [We — SIC-1i + [St—1 — SIC—2i + ' ° ' + [31+1 - 81]

< (k -— l)de

< 6 since (k — l)d is infinitely small.

Thus, (3,.) is strongly Cauchy in R.

Corollary 4.2 Let (s,.) be a sequence in R. Then, (3,.) converges strongly if and

only if (3,.“ -— 3,.) is a null sequence with respect to the order topology.

64

Corollary 4.3 The infinite series 23,10 a,. converges strongly in R if and only if the

sequence (an) is a null sequence in R.

Corollary 4.4 The series 23.10% converges strongly if and only if it converges ab-

00

solutely strongly, that is if and only if ano ]a,.] converges strongly.

Proof. We have, using Corollary 4.3, that

m
Z on converges strongly 4:) “113.10% = 0

n=0

(1) lim Ia,.] =0

111—100

CX)
4:) Z Ia,.I converges strongly.

n=0

The proofs of Lemma 4.5, Corollary 4.5 and Lemma 4.6 follow the same lines as

the those of the corresponding results in R; so we omit these proofs here.

Lemma 4.5 Let (s,.) and (t,.) be two sequences converging strongly in R to s and t,

respectively. Then, the sequence (3,. + t..) converges strongly to s + t.

Corollary 4.5 Let 2:0 a,. and 23,10 b,. be two infinite series converging strongly in

R to a and b, respectively. Then, the series Zfliomn + b,.) converges strongly to a+ b.

Lemma 4.6 Let (s,.) and (t,.) be two sequences in R converging strongly to s and t,

respectively. Then the sequence (s,.tn) converges strongly to st.

Again the non-Archimedicity of R gives us a nice result in Theorem 4.5, which
does not hold in R without the additional requirement that one of the series converge

absolutely.

65

Theorem 4.5 Let 2:004: and 2:10 b,. be two infinite series converging strongly
in R to a and b, respectively. Then, the series 2:20 c,., where c,. = 239:0 ajbn_j,

converges strongly to ob in R.

Proof. First, we show that 23;, c,. converges strongly in R. By Corollary 4.3,
it suffices to show that lim,._,°o c,. = 0. Since 23,10 a,. and 2:10 b,. converge strongly
in R, the sequences (an) and (b,.) are both strongly null in R. Hence, by Corollary

4.1, (an) and (b,.) are both bounded. Therefore, there exists B > 0 in R such that
|a,.I < B and |an < B for all n 2 0.

Let 6 > 0 in R be given. Then, there exists M E Z + such that

d:

<
lam] B

and |me < 3;; for all m 2 M.
Let N = 2M. Then, for all m 2 N, we have that

[cm] = Iaobm + albm_1+ - - - + a..._1b1 + ambOI
S Iaome +Ia1b,.._1I + ' ' ' + Iam_1b1I + Ia,..b0|

= laollbm|+lalllbm—1|+°--+lam—illb1|+|amllbol

de de de de
B-§+B-§+"'+'§B+-B—B
= (m+1)de

<6.

So, for all e > 0 in R, we can find N E Z+ such that |c,,.| < e for all m 2 N. Hence,

00

lim,._.°° c,. = 0 and thus 2,50 c.. converges strongly in R. It remains to show that
3,10 c" = ab.

Consider the sequence of partial sums (3%), where

32,. = Co+Cl+"'+an

66

2n
= 2 a,b,-
i+j=0
= (ao+a1+---+a,.)(bo+b1+-~+b,.)
+a0(b,.+1+~-+ b...) + a1(b,.+1+~-+ on...) + - -- + a.._1b,.+1

+b0(an+1 + ' ' ' + 0.2,.) + b1(an+1 + ' ' ' + a2n—l) + ' ° ' + bn—lan+1-
Note that

Ia0(bn+1 + ' ° ° + b2n) + 01(bn+1+'°'+ 5211—1) + - ' - + an—lbn+1I

S Ia0IIbn+1 + ' ' ° + b2nI+Ia1IIbn+l+ ' ' ' +b2n—1I+ ' ' ' +Ian—1IIbn+1I

l/\

B(Ibn+1 + ' ' ° + ()an +Ibn+1 + ' ° ' + b2n—1I'I' ° ' ‘ + lbn+ll)-
Let 6 > 0 in R be given. Since (b,.) is a null sequence, there exists N E Z + such that
|an < 51—6- for all n 2 N.
B
Hence, for all n _>_ N and for all p E Z+, we have that

Ibn+1+ ° ' ° + bn+pI S lbn+1l+ ' ° ' + Ibn+pi

< 9l£+...+.d_€
B B
E E
— (Pd)°§—(Wd);l§
< _.__
718’

where, in the last step, we made use of the fact that d is infinitely small and pn is an
integer, so that pnd < 1. Therefore, for all n 2 N, we have that
[010(bn+1 + ' ' ' + b2n) + a1(bn+1 + ' ' ' + b2n—1)+ ' ' ° + an—lbn+ll
e e e
B _ _ ... _ = ,
< (,nB + nB + + n.3,) 5
n times

Therefore,

£13.90 (00(bn+1 + ' ° ‘ + b2n) + a1(bn+1 + ‘ ' ' + b2n—l) 'l' ' ' ' + an—lbn+1) = O-

67

Similarly, we can show that
"113,10 (b0(a,.+1 + - - - + a...) + b1(a,.+1+~-+ 0.2,.-1) + - - - + b _.a,.+.) = 0.
Hence

”132032” = “Ii—glo((a0+al +---+a,.)(bo+b1+-~+b,.))+
”11.11010 (00(bn+1 + ' ' ' + b2n) + 01(bn+1 + ' ° ' + b2n—l) + ' ° ' + an—lbn+1)+
111530 (b0(an+1 + ' ° ' + a2n) + b1(0n+1 + ° ' ' + 02n—1)+“‘+ bn-lafl+1)

= nler:o((ao+al+'°'+an)(b0+b1+"'+bn))'

Let (A,.) and (3,.) denote the sequences of partial sums of 23,1011, and 23;.0 b,.,

respectively. Then A,. = a0 + a1 + - . . + an, lim,._.°o A,. = a; B,. = b0 + b. + - - ~ + b,.,

and lim,H00 B,. = b. Therefore,

..ILasen = 11.11:.“an
= (“1590 A,.) (13111.10 8,.) using Lemma 4.6

= ab.

Since 23:0 0,. converges strongly, it has one and only one limit. Hence

lim $2,. = lim 32,.“ = ab = lim s,.;
n—+00 n—ooo n—voo

SO

<1 (eo-

n=0 n=0

Lemma 4.7 Let (s,.) be a sequence in R converging strongly to 3. Assume there exist
j E Z+ and t E Q such that Ms...) = t for all m 2 j. Then Ms) = t; in particular,
3 79 0.

68

Proof. Let 6 = d‘“. Then there exists M E Z+ such that Ism—sI < e for all m 2 M.
Let N = max{j, M} Then we have that Is... — s] < e and Ms...) = t for all m _>_ N.

From [3” - s] < e, we have that
MsN - s) _>_ Me) = t +1 > t. (4.4)

But we know that

MsN — s) = min{MsN), Ms)} S t if Ms) 79 MsN). (4.5)
From Equation (4.4) and Equation (4.5), we infer that Ms) 2 MsN) = t.
Lemma 4.8 Let (s,.) be a sequence in R. Assume there exists j E Z+ such that
s,,. = 0 for all m 2 j. Then (3,.) converges strongly to 0.
Proof. Let c > 0 in R be given. Then

Ism—OI = IsmI =0<eforallm2j;

so lim,.._.oo 3,. = 0.

Combining the results of Lemma 4.7 and Lemma 4.8, we obtain the following

result.

Corollary 4.6 Let (s,.) be a sequence in R converging strongly to 3. Assume there
exists 3' E Z+ such that Ms...) = t for all m 2 j, where t can be either finite or

infinite. Then M3) = t.

Proof. If t is finite, then we are done, by Lemma 4.7. Otherwise, sm = 0 for all
m _>_ 3'. Then, using Lemma 4.8, we have that (s,.) converges strongly to s = 0. Hence

Ms) 2 00 = t.

The following lemma is a consequence of the fact that the topology induced on R

by the order topology in R is the discrete topology in R (see Theorem 3.8).

69

Lemma 4.9 Let (s,.) be a sequence in R whose members are purely real. Then (3,.)

is strongly Cauchy if and only if there exists j E Z+ such that s... = s,- for all m 2 j.

Proof. Let (s,.) be a strongly Cauchy sequence in R with purely real members. Then

there exists 3' E Z + such that
Is,,. — 81|< d for all m,l 2 j. (4.6)

Since the members of (s,.) are purely real, we obtain from Equation (4.6) that Ism —

3;] = 0 for all m,l 2 j. Hence, 3... = s,- for all m _>_ j.

Conversely, let (3,.) be a sequence in R whose members are purely real, and assume
there exists j E Z+ such that s... = s,- for all m 2 j. Then, given 6 > 0 in R, we have
that

Ism—sz]=0<eforallm,l2j,

and hence (s,.) is strongly Cauchy.

Corollary 4.7 Let (s,.) be a sequence in R whose members are purely real. Then
(8,.) converges strongly if and only if there exists j E Z+ such that s,,. = s, for all

ij.

As we see, the concept of strong convergence provides very nice properties, and
moreover strong convergence can be checked easily by virtue of Theorem 4.3 and
Corollary 4.3. However, for some applications it is not sufficient, and it is advanta-

geous to study a new kind of convergence.

4.2 Weak Convergence

70

Definition 4.4 A sequence (3,.) in R is said to be weakly convergent if and only if

there exists 3 E R such that for all e > 0 in R, there exists N E Z+ such that
”s... — s]]1/. < e for all m 2 N.
If that is the case, we call 3 the weak limit of the sequence (3...), and we write

3 = wk— lim 3...

n—voo

Lemma 4.10 Let (s,.) be a sequence in R that is weakly convergent. Then (3,.) has

exactly one weak limit in R.

Proof. Let s and t be two weak limits of (s,.) in R. We need to show that s = t.

Let c > 0 in R be given. There exists N E Z + such that
|Is,.. - sI|2/e < 6/2 and IIsm — tII2/e < 6/2 for all m 2 N.
It follows that

“3 — tliz/e = “3 — 3N + SN - t||2/e
S “SN — Sll2/e + “SN - tllz/e (4-7)

< e/2+e/2 =6.

Let r E Q be given. Choose 6 > 0 in R, small enough such that r < 2 / 6. Then, using
Equation (4.7), we have that

[(3 — t)[r]] < 6.

Since e can be chosen arbitrarily small in R, we deduce that ](s—t) [r]| = 0. Therefore,
(3 — t)[r] = 0.

Since r was an arbitrary rational number, we have that s — t = 0 and hence s = t.

 

hmmat

Proof. Let

kmma4.

“3Q. the-

Proof. Let

 

 

 

\tv. 1' \
”4?? 1371,] c

“398 7' <1

71

Lemma 4.11 Let r1, r2 E Q be such that r1 < r2. Then

”when S llxlle for all a: 6 R-

Proof. Let x E R be given. Then

lieu... = suptletqil : q s n} = mexnetqll =4 S 1‘2}
= max{max{lx[qll rq S r1},maX{|$[<Ill 1 7‘1 < ‘1 3 r2}}
= max{]]a:]],.l,max{Ix[q]] : r1 < q 3 r2}}

> llrvllri-

Lemma 4.12 Let (s,.) be a sequence in R converging weakly to 3. Then, for all

r E Q, the sequence (|Is,.]|,) converges in R to ”3”,.

Proof. Let r E Q and e > 0 in R be given. Let 61 > 0 in R be such that
e < min 1 e
1 1 + IrI’ '

Since (3,.) converges weakly to s in R, there exists N E Z + such that

]Is,,. - sIII/£1 < c. for all m 2 N.
Since r < 1 + [r] < 1/61, we have, using Lemma 4.11, that

]Is,,. — 3]], S ]Is,,. — sIIl/61 < 61 < e for all m 2 N.

Finally, using Equation (3.9), we obtain that

IIIsmII, — IIsII,I < e for all m 2 N,

which shows that (]Is,,II,) converges in R to “3”,.

Remark 4.3 The converse of Lemma 4.12 is not true.

72

Proof. Let (s,.) be the sequence in R whose terms are given by 32,. = 1 and 32).... =

—1. Then, for all n, we have that

1 if r 2 0
0 otherwise °

Ilsnllr = “In. ={

Hence, for all r E Q, Ilsnll, = [Ill], for all n; so the sequence (IIs,.II,.) converges in R

to ”1“,, for all r E Q. However, the sequence (3,.) does not converge weakly in R.

Theorem 4.6 (Convergence Criterion for Weak Convergence) Let the sequence
(3,.) converge weakly to the limit 3. Then, the sequence (s,.[q]) converges to qul in R,
for all q E Q, and the convergence is uniform on every subset of Q bounded above.
Let on the other hand (3,.) be regular, and let the sequence (s,.[q]) converge in R to

quI for all q E Q. Then (3,.) converges weakly in R to s.

Proof. Let (s,.) be a sequence in R converging weakly to 3. Let r E Q and e > 0 in

R be given. Let

, l
e < min ——,6
1 {1+ Ir] I
be given. Then

1/61 >1/e and 1/61 > 1+ Ir] > r.

Choose N E Z+ such that
]Is,,. — sII1/.1 < 61 for all m 2 N. (4.8)
Since r < 1/61, we have, using Lemma 4.11, that
]Is,,. — 3]], 3 ]Is,,. — slll/£1 for all m. (4.9)
It then follows from Equation (4.8) and Equation (4.9) that

]Is,,. — 3]], < 61 < e for all m 2 N.

LCD entta
that is, _r_

tonterges :

i" ' ifn
l: tlIanIIl

On the
m If IO 5:1,.
to Show the

‘1 conl

Only if ther

 

 

 

533998 at

is ['C’Illainet

“11:13..
“omdf. a:

73

Hence
Ism[q] — quII = I(s,,. — s)[q]l < e for all q _<_ r and for all m 2 N, (4.10)

which entails that, for all r E Q, and for all e > 0 in R, there exists N E Z”L such
that |s,..[r] — s[r]l < e for all m _>_ N. Therefore, for all r E Q, the sequence (s,.[r])
converges in R to sIr]. Moreover, from Equation (4.10), we see that the convergence

is uniform on the subset {q E Q, q S r} bounded above by r.

On the other hand, let (3,.) be a regular sequence in R and let (snlql) converge
in R to qu] for all q in Q, where s : Q —> R is a real-valued function on Q. We need
to show that s E R and that (s,.) converges weakly to 3. Let q E Q be given. Since
(s,.[q]) converges to quI, we have that quI = 0 if s,.[q] = 0 for all n. Thus, quI aé 0
only if there exists m E Z + such that s,,. [q] 74 0. Therefore, every support point of
s agrees at least with one support point of one member of the sequence, and hence
is contained in S = Uziosupp(s,.), which is left-finite since (3,.) is regular. Hence

supp(s) is a subset of S and is itself left-finite. So 3 E R.

Now let 6 > 0 in R be given, and let r in Q be such that r > l/e. We first show
that the sequence (s,.[q]) converges uniformly to qu] on {q E Q, q S r}. Since (3,.) is
regular, any point at which 3 can differ from any 8,. has to be in S. Thus, there are
only finitely many points to be studied below r, say ql, . . . ,qk. For j = 1,. . . ,k, find
Nj E Z+ such that

IsmIqj] — Squ‘ll < e for all m 2 Nj.
Let N = max{N,~ : j = 1,...,k}. Then IsmIq] — qu]I < e for all m 2 N and for
all q S r. In particular, [(sm — s)[q]l = ls,,.[q] - qull < e for all m 2 N and for all

q S 1/6. It follows that
”Sm - sill/e = max{|(s,,. — SilQl] 3‘] S 1/€}< e for all m 2 N,

which shows that (s,.) converges weakly to s.

74

Definition 4.5 Let (s,.) be a sequence in R. Then we say that (s,.) is weakly Cauchy

in R if and only if for all e > 0 in R, there exists N E Z+ such that

]Is,,. — sllll/e < e for all m,l _>_ N.

Lemma 4.13 Let (s,.) be a sequence in R that converges weakly. Then (3,.) is weakly

Cauchy.

Proof. Let s be the weak limit of (s,.) in R, and let 6 > O in R be given. Then there
exists N E Z+ such that ]Is,,. — sII2/. < 6/2 forall m 2 N. Let m,l 2 N be given.

Then,
“3m — 31li2/e = ”Sm + S — 3 — Silli/e
_<_ “Sm — SII2/e +||81- Sllz/e
< 6/2 + 6/2 = 6.
Using Lemma 4.11, we have that

“Sm _ SIill/e < ”Sm _ Slil2/e < €f0ra11m9l2 N)

from which we infer that (s,.) is weakly Cauchy.

The converse of Lemma 4.13 is not true, as the following theorem will show.
Theorem 4.7 R is not Cauchy complete with respect to the weak topology.

Proof. It suffices to find a sequence (3,.) in R that is weakly Cauchy but not weakly
convergent. Consider the sequence (3...), where

n d—j

Sn: f“.
E i

For all n E Z“, we have that supp(sn) = {-j : 1 S j S n} E .7. Hence 3,. E R for

alln.

let 6 ii.

_.
.a
'1'

if
:3
IV

....

Hence.

So. for all '

155211111

l I
'tth‘: I con

some I- E ,1]

Hence. the“

 

tn E
l (i
v ”.111:-

«lat

75

Let c in R be given. Choose N E 2‘" such that N > l/e. Then 5 > l/N 2 l/n
for all n _>_ N. Let m > Z 2 N be given. Then,
m -.-
Sm — 81 = Z L.
j=l+1 3
Hence,

re —su — 1 <.
m ll/E—l-l-l -

So, for all m,l 2 N, ]Is,,. — Still/e < 6. Hence (3,.) is weakly Cauchy.
Assume that (s,.) converges weakly to s in R. Then, by Theorem 4.6, the sequence
(s,.[q]) converges in R to s[q] for all q E Q. Let q E Z ’ be given. Then q = —k for

some k E Z +. Therefore,

=—% ifn_>_k

SM] = 34‘“ z] (i ifn < k '

Hence, the sequence (s,.[q]) converges in R to —%. So
1 _
quI=—E750forallqEZ,

from which we infer that supp(s) is not left-finite. This contradicts the assumption

that s E R. Hence (3,.) does not converge weakly in R.

Lemma 4.14 Let (s,.) be a sequence in R that is weakly Cauchy. Then (3,.) is weakly

bounded , that is there exists B > O in R such that Ilsnlll/p < B for all n.

Proof. Since (3,.) is weakly Cauchy, there exists N E Z + such that
”3,. — smII1< 1 for all n,m Z N.

In particular, we have that ]Is,. -— lell < 1 for all n 2 N, from which we obtain, us—
ing Equation (3.8), that |Is,.||1 < 1 + IlsNII1 for all n 2 N. Let 31 = 1 + IIlell.
Then l/Bl S 1, and hence I]s,.II1/B1 < Ilsnlll < 81 for alln 2 N. Let B2 =

76

max{||s,,||1/31 : n g N — 1} + 1, and let B = max{Bl,Bg}. Then, for all n 2 N, we
have that

llsnlll/B S llsnlll/B. < 31 S 3- (4-11)

Moreover, for all n S N — 1, we have that
“Saul/B S ”an1/31 < 32 _<_ B- (4.12)

Combining Equation (4.11) and Equation (4.12), we finally obtain that ||sn||1 /B < B

for all n.

Lemma 4.15 Let (s,.) be a sequence in R that is weakly Cauchy. Then for all r E Q,

there exists B, E R+ such that llsnll, < B, for all n.

Proof. Let r E Q be given. Then there exists N E Z + such that

“3,, — sm||1+|,l < for all n,m Z N.

1
1 + |r|
In particular,

1

Using Equation (3.8), we have that llsn||1+l,| < 1+||5Nll1+lrl for all n 2 N. Therefore,

using Lemma 4.11, we have that ”3,”, < l + “3N”1+|r| for all n 2 N. Let
Bl, —_- 1+ |ls~||1+m and 32,. = max{||s,,||1+l,l : n g N — 1} + 1,
and let B, = max{B1,,, B2,}. Then
”3,”, < 1+ ”SNHHIrI = Bl.r g B, for all n 2 N, and (4.13)

”3"”, < 32,, S B, for all n S N — 1. (4.14)

Combining Equation (4.13) and Equation (4.14), we finally obtain that ||sn||, < B,

for all n. This finishes the proof of the theorem.

77

The following lemma is a direct result of the fact that the topology induced on R

by the weak topology in R is the usual order topology in R (see Theorem 3.9).

Lemma 4.16 Let (3,) be a purely real sequence converging to s in R. Then, regarded
as a sequence in R, (3,) converges weakly to 3. 0n the other hand, let (3,.) be a
sequence in R with purely real members, converging weakly to 3. Then 3 is purely

real, and the sequence (3,) converges in R to s.

Proof. Let (3") be a purely real sequence converging to s in R. We now view (3,.)

as a sequence in R. Let c > 0 in R be given. There exists N E Z + such that
Isn—sl < efor alan N. (4.15)
Since, for all n, 3,, and s are purely real, we have that

llsn - Slll/e = |(8n - SW“

= Isn — sI. (4.16)

Combining Equation (4.15) and Equation (4.16), we have that “3,, — s||1/, < e for all

n 2 N. Hence (3,) converges weakly to s in R.

Now, let (3,) be a sequence in R with purely real members, converging weakly to
s E R. By Theorem 4.6, the sequence (sn[q]) converges to s[q] for all q E Q. Since
3,,[q] = O for all q 75 O and for all n, we have that s[q] = 0 for all q 79 0 and hence s
is purely real. To show that (s,.) converges to s in R, let 6 > O in R be given. Then
there exists N E Z+ such that Is, — s| = ”3,, — s||1/,E < e for all n 2 N, which finishes

the proof of the theorem.

Lemma 4.17 Let (s,.) and (t,.) be two sequences in R converging weakly to s and t,

respectively. Then the sequence (3,, + tn) converges weakly to s + t.

 

78

Proof. Let 6 > 0 be given in R. Then there exists N E Z + such that
”3,, — s||;_>/,E < 6/2 and Mt, — t||2/,E < 6/2 for all n 2 N.
Therefore, for all n 2 N, we have that

“(Sn + tn) - (8 +t)||1/e S “(Sn + tn) - (8 + t)llz/e

|/\

“3n — SH2/e + “tn - tH2/e

< e/2+e/2 = 6.

Hence (3,, + tn) converges weakly to s + t.

Corollary 4.8 Let 2:10 on and 23;.0 bn be two infinite series in R converging weakly

to a and b, respectively. Then the infinite series 2:0(01: + bn) converges weakly to

a+b.

Proof. Let (An) and (8,) be the sequences of partial sums of 2:10 an and 23:0 bu,
respectively. Then (A,.) and (Ba) converge weakly to a and b, respectively. Therefore,
by Lemma 4.17, (A,. + Bn) converges weakly to a + b. But (A,. + B") is the sequence
of partial sums of 22:0(an + bu). Hence 33:0(an + bn) converges weakly to a + b.

We can thus write

2(an+bn)=a+b=:an+2bn.
n=0

n=0 n=0

Theorem 4.8 Let (3,) and (t,.) be two regular sequences in R converging weakly to

s and t, respectively. Then the sequence (sntn) converges weakly to st.

Proof. Since (3") and (t,.) are both regular, so is (sntn), by Lemma 4.1. To show
that (s,.tn) converges weakly to st, it remains to show that the sequence ((sntn)[q])

converges in R to (st)[q] for all q E Q, using Theorem 4.6. Let A = Uf=osupp(an)

79

and B = Uziosuppwn). Then A,B E .77. Let q E Q be given. Then, for all n, we

have that

(Sntn) [Q] = 2 3n [q1]tn [‘12]- (4-17)

01+¢n=¢1
«621.9268

Since A and B are left-finite, only finitely many terms contribute to the sum in

Equation (4.17); and we have that

JLHgJSntn) [q] = lim 2 3“ [Q1]tn[q2]

n-—+oo
<11 +¢n =q
91621.92 68

= 2 (gig; (8n[qlltn[q2]))

91+92=9
QIEAvTIEB

= 2 ((31,220 Slell) (JgngotdqflD

91+42=9
QIEAIQQEB

= Z (s[ql]t[q2])

qi+qa=q
GIGAoqzéB

= (st)[ql-

This finishes the proof of the theorem.

Theorem 4. 9 If the series Ere an and 23:0 bn are regular, 2:0 0., converges ab-

solutely weakly to a, and 2” bn converges weakly to b, then ELM, en, where on =

n=0

Zj_0 ajb n_j, converges weakly to ob.

Proof. Let (A,.), (8,), and (0,.) be the sequences of partial sums of 2:10 on, 2:10 b",
and 2310 c", respectively. Then (A,.) and (3,.) are both regular, (An) converges
absolutely weakly to a and (Ba) converges weakly to b. Since (A,.) and (Ba) are both

regular, so is (0,). It remains to show that (Cn[q]) converges in R to (ob) [q] for all
q E Q.

80

Since (An) converges absolutely weakly to a, (A,.[t]) converges absolutely in R
to a[t] for all t E Q. Similarly, (Bn[t]) converges in R to b[t] for all t E Q. Let
A: U§°__.0supp(a,,) and B: U;‘,°__Osupp( bu), and let q E Q be given. Then

0an = (z c...)[q1= i emu]

m=0 m=0

= i (m (ajbm m-j Md)

m: '=0

0

= 2": in: Z “1' [qllbm—j [(12]

m= j=0 q1+q2=q
916144263

O

= Z (Z Z ailQllbm_,-[q2]) because of regularity

qi +q2 = q m=0j=0
ql E Afin E B

= 2 (En: (iajlqllbm—qul» -

91 +q2 = q m=0 j=0
916 4.92 6 3

Since 2 3100—0 a,.[q1] converges absolutely to a[ql] and since 2310 bn[q2] converges to b[Q2],

7.1390 (2:20 (i aj[‘11lbm—j[(12l))

exists in R and is equal to a[q1]b[q2]. Since the sum over the q’s is finite because of

we have that

left-finiteness of A and B, we have also that

.3122. 2 (2(iajlqllbm-.[q21))

Qi+qz=q m=0 j=0
vi EA.q2EB

81

exists in R and is equal to

2: (7315.10 (i=0 (:1 “j [q1]bm_j [42]) )) -

916144268

Hence lim,H00 0,, [q] exists in R and we have that

”1133.0 cnrq] = z a[qilblfhl =(ab)[q1.

q1+92=q
<11 €A.92€B

Since (0,) is regular and since lim,Moo Cn[q] = (ab) [q] for all q E Q, (Cn) converges

weakly in R to ob. Therefore, 233:0 c,l converges weakly to ab, and we can write
xen=ab=<zan><z~>
n=0 n=0 n=0

The relationship between strong convergence and weak convergence is provided

by the following theorem.
Theorem 4.10 Strong convergence implies weak convergence to the same limit.

Proof. Let (3") be a sequence in R converging strongly to 5. Then, by Lemma 4.4,
we have that (s,.) is regular. To show that (s,.) converges weakly to s, it suffices by
Theorem 4.6 to show that the sequence (s,.[q]) converges in R to s[q] for all q E Q.
So let q E Q be given. Since (3,) converges strongly to s in R, there exists N, E Z +
such that

|sm — s| < qu for all m 2 Nq.

Thus,

sm[q] = s[q] for all m 2 Nq,

which entails that the sequence (s,,[q]) converges in R to s[q].

 

82

4.3 Power Series

We now discuss a very important class of sequences, namely, the power series. We
first study general criteria for power series with R coefficients to converge strongly
or weakly. Once their convergence properties are established, they will allow the
extension of many important real functions, and they will also provide the key for
an exhaustive study of differentiability of all functions that can be represented on a
computer (see Chapter 7; also see [38, 39, 40]). Also based on our knowledge of the
convergence properties of power series with R coefficients, we will be able to study in
Chapter 6 a large class of functions which will prove to have all the nice smoothness
properties that real power series have in R; (see also [41, 43]). We begin our discussion

of power series with an observation.

Lemma 4.18 Let M E f be given. Define
M2 = {q1 + +qn : n E Z+, and q1,...,q,, E M};

then M): E f if and only if min(M) Z 0.

Proof. Let g = min(M). First assume that g < 0. Clearly, all multiples of g are in
M2. In other words, Mg contains infinitely many elements smaller than zero and is
therefore not left-finite. Now assume that g 2 0. For g = O, we start the discussion
by considering M = M \ {O}, which has a minimum greater than zero. But since M
differs from M only by containing zero, and since inclusion of zero does not change
a sum, we obviously have that M2 = Mg. It therefore sufices to consider the case
when 9 > 0. Now let r E Q be given; we show that there are only finitely many
elements in M2 that are smaller than r. Since all elements in M): are greater than or
equal to the minimum g, the property holds for r < 9. Now let r 2 g be given, and let

n = integer(r / g) be the greatest integer less than or equal to r / g. Let q < r in Mg be

mg. IF-

 

rlF—- u

83

given. Then at most n terms can sum up to q, since any sum with more than n terms
exceeds r and thus q. Furthermore, the sum can contain only finitely many different
elements of M, namely those below r. But this means that there are only finitely
many ways of forming sums, and thus only finitely many results of summations below

7‘.

Corollary 4.9 The sequence (50") is regular if and only if A(a:) Z 0.

Let (an) be a sequence in R. Then the sequences (anrn) and (2;;0 aJ-xj) are

regular if (an) is regular and A(:r) Z 0.

Proof. First observe that the set UflesuppCr“) is identical with the set Mg in the
previous lemma if we set M = supp(x). This is left-finite if and only if supp(x) has

a minimum greater than or equal to zero; that is if and only if Mr) 2 O.

The second part is an immediate consequence of Lemma 4.1, which asserts that

the product of regular sequences is regular.

Theorem 4.11 (Strong Convergence Criterion for Power Series) Let (an) be

a sequence in R, and let

A0 = — lim inf (M70122) = limsup (:1?) in R.

n—mo

Let 9:0 E R be fixed and let a: E R be given. Then the power series 23,10 an(:t — x0)"

converges strongly in R if /\(a: - x0) > A0 and is strongly divergent if Ma: — .730) < A0

or if M2: — 2:0) = A0 and —)\(a,,)/n > A0 for infinitely many n.

Proof. First assume that /\(:z: - x0) > A0. To show that 23,10 an(a: —- x0)" converges
strongly in R, using Corollary 4.3, it suffices to show that the sequence (an(a: — x0)")

is a null sequence with respect to the order topology. Since Mr — x0) > A0, there

 

84

exists t > O in Q such that

 

/\(:r: - 2:0) - t > A0 =limsup(—A1(1an)) .

n—+oo

Hence there exists N E Z + such that

Man)

/\(x—$0)-t>;7z—foralanN.

Thus,

A(a,,(a: — 330)") = Mon) + nA(:c — :co) > nt for all n 2 N.

Since t > 0, we obtain that (an(a: — 300)") is a null sequence with respect to the order

topology.

Now assume that Ma: — 9:0) < A0. To show that 2.3100411? — x0)" is strongly
divergent in R, it suffices to show that the sequence (an(x — 330)") is not a null
sequence with respect to the order t0pology. Since A(a: — 330) < A0, for all N E Z+

there exists n > N such that

/\(a: — $0) < —/\1(za,,).

 

Hence, for all N E Z +, there exists n > N such that
/\(a,,(a: — 230)") < 0,
which entails that the sequence (an(x — 1100)") is not a null sequence with respect to

the order topology.

Finally, assume that /\(a: — $0) = A0 and —)\(a,,)/n > A0 for infinitely many n.

Then for all N E 2“, there exists n > N such that

fist—n)- > A0 = Mas—3:0).

Thus, for each N E Z+, there exists n > N such that /\(a,,(x — 120)") < 0. Therefore,

the sequence (an(x — x0)") is not a null sequence with respect to the order topology;

85
and hence 2:10 an(a: — x0)” is strongly divergent in R. This finishes the proof of the
theorem.

The following two examples show that for the case when /\(a: — x0) = A0 and
—)\(a,,)/n 2 A0 for only finitely many n, the series 2:10 an(a: — 2:0)" can either
converge or diverge strongly. For this case, Theorem 4.12 provides a test for weak

convergence.

Example 4.1 For each n 2 0, let an = d; and let $0 = 0 and a: = 1.

Then

 

n—-+oo n

A0 = limsup (—/\(a,.)) =1imsup (——) = 0 = A(:z:).
Moreover, we have that

—).(a,,) = #11; < A0 for all n 2 0;

 

n

and 2:10 aux" = 3,10 d is strongly divergent in R.

Example 4.2 For each n, let qn E Q be such that \/r_i/2 < q" < fl, let on = d“;

and letxo=0 andx= 1.

Then
A0 = lim sup (11811)) = lim sup (_%) = O = /\($).

n—boo

Moreover, we have that

M=_fi<0=)\oforalln20;
n n

and 2 :°°,,_o aux" = 22:0 (19" converges strongly in R since the sequence (dq") is a null

sequence with respect to the order topology.

86

Theorem 4.12 (Weak Convergence Criterion for Power Series) Let (an) be a
sequence in R, and let

A0 = - lim inf (M3)) = lim sup (#> in R.

”"’°° n—roo

 

Let mo E R be fixed, and let x E R be such that /\($ — :30) = A0. For each n 2 0,
let b, = and”’\°. Suppose that the sequence (bu) is regular and write U§°=03upp(bn) =
{q1,q2,...}; with q,, < qu if jl < jg. For each 17., write b,, = 23:, bnjd‘b', where

1
1‘ = .
sup {limsup.._... Ib.,I1/" zj 2 1}

 

Then 2” an(a: — 2:0)" converges absolutely weakly in R if ](x — ro)[)\0]| < r and is

71:0

weakly divergent in R if |(a: - $0)[/\0]| > r.

Proof. Letting y = d"‘°(.r — 220), we obtain that

 

n—ooo

—). b,
/\(y) = O = limsup( T: )), and an(:c — x0)" = by" for all n 2 0.
So without loss of generality, we may assume that
320:0; AO=O=A(r); andbn=anforalln20.

Let X = 32(3); then X 75 0. First assume that IX] < r.
First Claim: For all j 2 1, we have that 2:10 anj X " converges in R.

Proof of the first claim: Since IX | < r, we have that

1
m > sup{lirln_+s°1.ip|anj|1/”:j 2 1};

and hence

1
— > limsupIan.|1/" for allj Z 1.
|X| n—ooo ’

 

87

Thus,
1

lim sum...» Ian - I”

 

|X|<

Hence 2°10 an. X " converges in R for all j > 1.

nforallj_>_1.

Second claim: For all j _>_ 1, 23:0 anjx" converges weakly in R.

Proof of the second claim: Let j Z 1 be given. For each n, let A,.j(a: 1:): 2:0 {1,-1.39. So
we need to show that the sequence (A,.). (93)),20 is weakly convergent. Using Corollary
4.9, the sequence is regular since Mr) 2 O and since the sequence (aij) is purely real
and hence regular. Thus, it suffices to show that the sequence (Am (:13) [t]) converges

in R for all t E Q. Let s = .7: — X. If s = 0, then we are done. So we may assume that '

 

s 79 0. Let t E Q be given; and choose m E Z+ such that m/\(s) > t. Then (X + s)" E

evaluated at t yields:
«X + a") [t] = (2s‘mf —'—,‘,,,.X"-') [t]

min{m,n}

= Z sn_’[t]-(———- ”WW“

1:0

For the last equality, we used the fact that s vanishes at t for l > m. So we get the

following chain of inequalities for any V2 > V1 > m:

 

 

           

 

 

 

 

p2 min{m,n}

"=2” |a.,.(X+s)"[t1| = 2 |a.| _ 2snf‘lt1-(—— 'muxr
s W1 [:0 s[tllm -——,—_).,.|XI"‘l
< (in) Sim mm) (f: la nmlxl”'"‘)
— (=0 “ n=V1 "j .

Note that the right sum contains only real terms. As IX | < r, the series converges; the
additional factor n'" does not influence convergence since hm...”o {Vnm = 1. As the
left hand term does not depend on u] and V2, we therefore obtain absolute convergence

at t. This finishes the proof of the second claim.

88

Third claim: 211—O aux" converges weakly in R.

Proof of the third claim: By the result of the second claim, we have that 272-0 an 2:"

converges weakly in R for all j Z 1. For each j, let fj(:c )= Zn—O anjr"; then
A(f,-(a:)) 2 0 for alljZ 1.

Thus 2.211 qu' fj (:13) converges strongly (and hence weakly) in R. Now let t E Q be

given. Then there exists m E Z+ such that q,- > t for all j 2 m. Thus,

(2441mm) [t1 = 2<dri< >>[t1=2( 2 drawers)

j=l t1+t2=t

= i( Z dqi[t1]f,-(a:)[t2]) =2: 2 dailt1](ioanjmn) [t2]

j=1 t1+t2=t = t1+t2=t

= 2 2 dq’lt112amrc"[t21= 2%( 2 dq’ltllx"lt21)

j=l t1+t2=t n=0j-1 t1+t2=t
CD 00 00 (X3
= Z Zuni Z dq’ [t1]:1:"[t2] = Z 20'": (dqjxn) [t]
n=0 j=l t1+t2=t 71:0 j=l

= (i (in-Tn) [t].

n=0

This is true for all t E Q. Thus, 2:10 aux" converges weakly to 23:1 qu' f,- (:13).

Now assume that IX | > r. Then

1
m < sup {limsuplaijI/n :j Z 1}.

Hence there exists jo E Z + such that

1
— < limsu . V".
|X| nsmpla'bo'

 

89

Thus,
1 0
lim supngoo |a,,.o |1/"’

 

le>

and hence 2:10 anmX" diverges in R. Therefore, (2,20 ans: ") [qjo] diverges in R; and

hence 2°10 aux“ is weakly divergent in R.

Corollary 4.10 (Power Series with Purely Real Coefficients) Let 2,30%
an E R, be a power series with classical radius of convergence equal to 1). Let :1: E R,
and let An(:1:) = 2:20 airi E R. Then, for |:1:| < 17 and |:1:| at 17, the sequence (An(:c))

convergesabsolutely weakly. We define the limit to be the continuation of the power

series on R.

4.4 Transcendental Functions

Using Corollary 4.10, we can now extend real functions representable by power series

to the new field R.

Definition 4.6 (The Functions Exp, Cos, Sin, Cosh, and Sinh) By Corollary
4.10, the series

2%?2113—211.15—7....2735.’ «1112......

converge absolutely weakly in R for any a: E R, at most finite in absolute value. For

any such :11, define

expm = 2r?»

 

008(3) = Z(_1)n(2n)1;

sin(x) = Z(-1)nm;

:5!

 

90

 

h 00 $211
cos (x) -— g (271)!)
' oo x2n+1
s1nh(:c) — mom.

Remark 4.4 It follows from Definition 4.6 that for any a: E R, at most finite in
absolute value, cos(x) and cosh(:r) are even functions, while sin(x) and sinh(:1:) are

odd functions of :13.

Theorem 4.13 (Addition Theorem for the Exponential Function) Let 21:1, :32

E R be at most finite in absolute value. Then

€XP($1)9XP($2) = eXP($1 + 1132)-

Proof. Since 2210 a? and 2230211 both converge absolutely weakly in R for any

x1 and $2, at most finite in absolute value, we have by Theorem 4. 9 that E210 c,
converges weakly in R to ( 2°___0 a?) ( 22:0 1,), where
(”(11-1)
cu:
j=0 1311M-
Hence,

-..... = (<21) (:1)=2(2“—1)

..:o ,-=oJ'!(n J)!

0° 1 n n! - ("_1) 0° 1 n
= ”(En—371’” ’)=Za<11+12>

n=0 n! j=0 n=0
exp(:1:1 + 51:2).

Corollary 4.11 Let :1: E R be at most finite in absolute value. Then

exp(:1:) ~exp(—a:) = 1.

 

‘ if

91

Thus for any such :13,

 

exp(m) =74 0 and expl(:c) == exp(—:z:).

Corollary 4.12 Let :1: E R be at most finite in absolute value. Then

exp(sc) z ex, where X = 32(23).

Proof. Write a: = X + 3. Then by Theorem 4.13,

exp(x) = exp(X) exp(s) = ex exp(s).

Since Isl is infinitely small, exp(s)~ ~ 1. Thus exp(x)~ ex since ex 75 0.

Theorem 4.14 (Addition Theorems for Cosine and Sine) Let $1,122 E R be

at most finite in absolute value. Then

cos(zcl :1: x2) = cos($1)cos(x2) IF sin(x1)sin(a:2), and (4.18)

sin(zcl :t 232) = sin(ml)cos(:1:2) :i: cos(zl) sin(zg). (4.19)

Proof. Using the definitions of the sine and cosine functions in Definition 4.6, we

have that

cos(xl) cos(zcg) IF sin(x1)sin(x:)

= (2<-—1>n3;,,,)(>_:<-1>n3:3,:)

n—O '

(§‘"22:12—.)(§< "(T—.5312)

 

 

n 321' 211—21' 7; 2j+1 x2n-2j+l

 

 

n ‘52 $1 2
= 1 '¥ .
.;( ) 59]? '-(Zn 22')‘ 2(1) n§(2j+l)l(2n—Zj+l)l
n 23' 2n—2j n—l 2j+l 2n—2j-1

 

 

§(-l)n,§2(2j)'(2n- 23M; 31:3 )n—,=0(2j+1)!(2n— 2j—1)!

q

 

I-m-r—lu—IAA' '_‘

92

$2j x2n- -2J' oo $2j+1 x2n—2j—l

= n;(—l)n.}:(2j)l2(2n—2j)l i;(-1)":§<2j+1),(2n2_2j_1),

 

 

 

 

n 27' 2n—2j n—l 2j+1 m2n—2j—1 )

= 1+2( 1)(22(21)!(2n2—2J')'if3=312(211+1)’(2"2—2j‘1)!

 

 

_ oo — n 111:2]. (i232 )2n—2J' n—l fo-H (i172 )2n—2j—l
_ 1+2; 1) (§,(2J)'(Zn- 2J')‘ =0(2J'+1)'(2n- 22J'-1)')

271 ill: (i$2)2n_k

= 1+231:12! (222—12)!

n=l k=0

=1+Z(1(($—1—2—,——in32)% ,

 

__ ”(Tli—L2)%
_ :(_1) (2n)!

= COS 0:($1 i 1:2),
which proves Equation (4.18). Similarly, we show that Equation (4.19) holds.

Corollary 4.13 Let .’L‘ in R be at most finite in absolute value, let X = 9201:), and
lets=x-—X. Then

cos(x) z { 08(X) if cos(X) 75 O
‘S'Si9n(Sin(X)) if cos(X) =0 ’

where

. 1 iY>O
“9"(Y)={ —1 ilY<0 '

Proof. By Theorem 4.14, we have that
cos(cc) = cos(X) cos(s) — sin(X) sin(s).

Suppose cos(X) aé 0; then

cos(x)=cos(X)(1+Z:(-)1 (-:—23,)— s1n(X) (2(1) T—zfi),)zcos(xy

n-O

93

Now suppose that cos(X) = 0; then sin(X) = sign(sin(X)), and hence

cos(x) = — sin(s)sign(sin(X)) = — (ix—1 1:31)?) sign(sin(X))

":0 )n (2n + 1
z -s - sign(sin(X)).

Corollary 4.14 Let 11:, X, and s be as in Corollary 4.13. Then

sin(x) z { sin(X) if sin(X) ¢ 0
S'sign(COS(X)) if sin(X) = 0 -

Proof. By Theorem 4.14, we have that
sin(s)) = sin(X) cos(s) + cos(X) sin(s).
Suppose sin(X) 75 0; then

. . 00 71 S21; 00 n s2n+l N .
$1n(a:) = sm(X) (1+ nay-1) W) + cos(X) (g(—1) m) ~ s1n(X).
Now suppose that sin(X) = 0; then cos(X) = sign(cos(X)), and hence

sin(x) = sin(s)sign(cos(X)) z s 2 sign(cos(X)).

Corollary 4.15 Let a: E R be at most finite in absolute value. Then
|sin(:c)| S |J:|.

Moreover, equality holds only if a: = 0.

Proof. Let X = like), and let 3 = a: — X. It suffices to show that
sin(r) g :1: for O 5 X S 1r/2,
and that equality holds only if a: = 0. Suppose X = 0. Then

3
sin(x) = sin(s) z s — g,- S 3; thus sin(m) = sin(s) S s = :13. (4.20)

 

94

The equality holds in Equation (4.20) only if a: = s = 0. Now suppose that 0 < X 3
7r / 2. Then by Corollary 4.14,

sin(a‘) z sin(X) < X z :13. (4.21)
Since X — sin(X) is finite, Equation (4.21) entails that sin(at) < 1:.
Corollary 4.16 Let cc E R be at most finite in absolute value. Then

cosz(:1:) + sin2(:v) = 1.

Proof. Using Theorem 4.14, we have that

 

cos2(:1:) + sin2(:1:) = cos(r) cos(x) + sin(x) sin(x) L.
= cos(x — :13) = cos(O)

=1.

Using the results of Theorem 4.14 and Corollary 4.16, we readily obtain the fol-

lowing two corollaries.

Corollary 4.17 Let :1: E R be at most finite in absolute value. Then

cos(Zx) = cosz(r) — sin2(a:) = 2cos2(:r) — 1 = 1 — 2sin2(a:), and

sin(2x) == 28in(:z:)cos(a:).

Corollary 4.18 Let :1: E R be at most finite in absolute value. Then

_ 1 + cos(2x)

1 — cos(Zx)
2 ———.

, and sin2(a:) = 2

cos2 (:13)

Definition 4.7 For any a: in R, at most finite in absolute value and satisfying

cos(x) aé 0, we define
sin(x)
cos(zc)

 

tan(a:) =

95

Definition 4.8 For any a: in R, at most finite in absolute value and satisfying

sin(x) 79 0, we define

Corollary 4.19 Leta: E R be at most finite in absolute value and satisfy sin(x) cos(x) 75
0. Then

 

cot(a:) - 1 f

_ tan(:z:)'

Corollary 4.20 tan(a:) and cot(a:) are both odd functions of a).

 

Corollary 4.21 Let $1 and 2:2 in R be such that tan(:z:1), tan(xg), and tan(a:1+:1:2) all
exist in R (i e. lel and |x2| are both at most finite, and cos(xl)cos(:cg) cos(zr1+x2) #

0). Then
tan(:z:1) + tan(:z:2)
1 — tan(a:1)tan(a:2)'

 

tan(:rl + x2) 2

Corollary 4.22 Let 2:1 and 2:2 in R be such that cot(a:1), cot(:1:2), and cot(x1+a:2) all
exist in R (i.e. lel and ngl are both at most finite, and sin($1)sin(1:2)sin(xl +232) ¢

0). Then
cot(a:1) cot(a:2) — 1
C0t($1) + C0t($2) .

 

C0t($1 + .732) =

Lemma 4.19 Let :1: E R be at most finite in absolute value. Then

and sinh(x) = exp(x) —2exp(—:c).

exp(x) + exp(—a:)
2

 

 

cosh(x) =

Proof. Using Definition 4.6, we have that

exP(-’I=) + “IX—”3) = g (i x" + i $.73?)

—v
2 n=0 7" n=0

 

— 1 W $fl+(_$)fl
22 n!

n=0

96

00 2k

=2L

k=0 (2k)!
= cosh(x).

Similarly, we can show that the second equality holds.

Corollary 4.23 Let 3:1, 2:2 E R be at most finite in absolute value. Then

cosh(:vl :1: 2:2) = cosh(:cl) cosh(:c2) :l: sinh(a:1) Sll’lh($2), and

sinh(:1:1 :1: x2) = sinh(:1:1) cosh(x2) :l: cosh(a:1) sinh(a:2).
Proof. Using the result of Lemma 4.23, we have that
cosh(a:1) cosh(:c2) + sinh(a:1) sinh(a:2)

exp(xl) + exp(—:r1)exp(x2) + exp(—:r2) +
2 2

 

exp(xl) - exp(—x1)exp(a:2) — exp(—a:2)
2 2

 

 

exp(zl + 2:2) + exp(xl — x2) + exp(—x1 + $2) + eXp(--’131 - 1132) +
4

eXp(xl + $2) — exp(xl -' $2) " exp(—a:1 + $2) + exp(—x1 " $2)
4

 

exP(-731 + $2) + exp(—:1:1 — 932)
2

 

= cosh(:1:1 + 3:2).
Similarly, we can show that
cosh(a:1 — 11:2) = cosh(:z:1) cosh(a:2) — sinh(x1) sinh(a:2), and

sinh(a:1 :l: 2:2) = sinh(a:1) cosh(:r2) i cosh(2:1) sinh(:1:2).

Corollary 4.24 Let :1: in R be at most finite in absolute value, let X = §R(x), and

let 3 = a: — X. Then the following are true

‘ Ira '
.-

 

97

cosh(rc) z cosh(X).

sinh(a:) z { :inh(X) 3% :3 .

cosh2(a:) -— sinh2(:r) = 1.
cosh(2x) = cosh2(:z:) + sinh2(x) = 2cosh2(:c) — 1 = 23inh2(a:) + 1, and
sinh(2:r) = 28inh(:z:) cosh(sc).

cosh2(a:) = W, and sinh2(:c) = W.

2 2

Proof. The proofs of the statements above are similar to those of the corresponding

results about the nonhyperbolic functions; and we will omit the details here.

 

Chapter 5

Calculus on R

In this chapter, we begin with a review of topological continuity and differentiabil-
ity. We show that, like in R, the family of topologically continuous or differentiable
functions at a point or on a domain is closed under addition, multiplication and com—
position. We also show that if the derivative exists, it must vanish at a local maximum
or minimum. However, we show with examples that, unlike in R, a topologically con-
tinuous or differentiable function on a closed interval need not be bounded or satisfy
any of the common theorems of real calculus. We then review continuity and differ-
entiability, based on the derivate concept [10]. We show that the class of continuous
or differentiable functions on a given interval of R is again closed under operations
on functions. We develop a tool for easily checking the differentiability of functions
and we finally use the new smoothness criteria to study a large class of functions
for which we generalize the intermediate value theorem in [5] and prove an inverse
function theorem. We study infinitely often differentiable functions, convergence of
their Taylor series and show that power series can be reexpanded around any point

of their domain of convergence.

98

 

99

5.1 Topological Continuity and Topological Differ-
entiability

Notation 5.1 Let a < b be given in R. By I (a, b), we will denote any one of the the
intervals [a, b], (a, b], [a, b) or (a, b).

Definition 5.1 Let D C R, and let f : D —+ R. Then we say that f is topologically
continuous at x0 E D if and only if for all e > O in R there exists 6 > 0 in R such
that

xEDand ]x—x0|<6=>|f(x)—f(xo)|<e.

Definition 5.2 Let D C R, and let f : D —> R. Then we say that f is topologically

continuous on D if and only if f is topologically continuous at x for all x E D.

The following example shows that, contrary to the real case, a function topologi-

cally continuous on a closed interval [a, b] of R need not be bounded on [a, b].

Example 5.1 Let f : [0,1] —* R be given by

ml yogx<d
f(x)= d‘1/’\("‘) idex<<1.
1 if x ~ 1
Then f is topologically continuous on [0,1]: Let x E [0,1] and let 6 > O in R be
given. First assume that O S x < d. Let 6 = (d — x) /2. Then 6 > O and for all

yE[O,1] satisfying Iy—xl<6,wehavethat0_<_y<x<dorOSx<y<x+6=
(d+x)/2<d. Thus f(y)=d‘1; and hence |f(y)—f(x)|=0<e.

Now assume that d S x << 1. Let 6 = d - x. Then for all y E [0, 1] satisfying

Iy — x] < 6, we have that A (y) = A (x). Thus, either

0<y<d§xandA(x)=)\(y)=1,

3‘."

 

100

in which case
|f(y) — f(x)l =ld-1— d‘ll = 0 < e;

01'

dgx,y<< 1 and A(x)=/\(y),

in which case

If (y) — f (x)l = let—W”) — d-W = 0 < ..

 

Finally, assume that x ~ 1. Let 6 = d - 1:. Then for all y E [0,1] satisfying
ly — x] < 6, we have that y ~ 1. Thus f(y)=1= f(x); and hence |f (y) — f(x)| =
O < 6.

Next we show that f is not bounded on [0,1]: Let M > 0 be given in R. Let

(1”? if MM) 2 —1
1: = .
d—‘fll-A M if ,\(M) < —1

 

Thus
[\(x)={% ifA(M)2—1
1-A1(M) E (0,%) HQ ifA(M) < —1 '
Hence
_ d—2 if,\(M) _>_ —1
”(If)“ — { dMMl‘l ifA(M) < _1

>> M.

Thus, for all M > 0 in R, there exists x E [0,1] such that If (x)| > M. So f is not
bounded on [0,1].

Lemma 5.1 Let D C R and let f : D —; R. Then f is topologically continuous at
x0 E D if and only if for any sequence (xn) in D that converges strongly to x0, the

sequence (f (xn)) converges strongly to f (x0).

15'

 

101

Proof. Suppose f is topologically continuous at am, and let (xn) be a sequence in D
that converges strongly to x0. Let 6 > 0 be given in R. There exists 6 > 0 in R such
that

xEDand Ix—xo|<6=>|f(x)—f(xo)|<e.

Since (xn) converges strongly to x0, there exists N E Z + such that
|xn—xol <6for alan N.

Thus,
”(5%) - f(x0)| < e for all n 2 N.

Hence the sequence (f (xn)) converges strongly to f (x0).

Now suppose f is not topologically continuous at x0. Then there exists 60 > O
in R such that for all 6 > 0 in R there exists x E D such that Ix — xol < 6 but

If (x) — f (xo)| > 60. In particular, for all n E Z +, there exists xn E D such that
Ixn — xol < d" and |f (xn) -— f(xo)| > 60.

Hence (xn) is a sequence in D that converges strongly to x0; but the sequence (f (xn))

does not converge strongly to f (x0).

Theorem 5.1 Let D C R, let f, g : D —> R be topologically continuous at x0 E D,

and let a E R be given. Then (f + ag) and (f - g) are topologically continuous at x0.

Proof. Let (xn) be a sequence in D that converges strongly to 1:0. By Lemma 5.1, we
have that the sequences (f (xn)) and (g (xn)) converge strongly to f (x0) and 9 (x0),

respectively. For all n 2 1, we have that

(f + 0‘9) ($71) = f (3311) + 09051:) -

 

102

Using the results of Lemma 4.5 and Lemma 4.6, the sequence (( f + ag) (xn)) con-
verges strongly to f(xo) + ag(xo) = (f+ag) (x0). By Lemma 5.1, (f +ag) is

topologically continuous at x0.

Also, for all n 2 1, we have that

(f ‘ 9) (xn) = f (3n)9($n)
= (f (3n) - f($o)) (g(l‘n) - 9080))
+f ($0) (9 (In) — 9 (130))
+9030) (f (mu) — f (130))
+f ($090130)-

Thus, the sequence (( f - g) (xn)) converges strongly to f (x0) g(xo) = (f- 9) (x0).

Again, by Lemma 5.1, (f - g) is topologically continuous at x0.

Corollary 5.1 Let D C R, let f, g : D -—» R be topologically continuous on D, and

let or E R be given. Then (f + 09) and (f - g) are topologically continuous on D.

Theorem 5.2 Let Df,Dg C R and let f : Df —» R and g : D9 —+ R be such that
f (Df) C D9, f is topologically continuous at x0 E Df and g topologically continuous
at f(xo). Then 9 o f : Df -—> R, given by (gof) (x) = g(f (x)), is topologically

continuous at x0.

Proof. Let (xn) be a sequence in Df that converges strongly to x0. Since f is topo-
logically continuous at x0, the sequence (f (xn)) converges strongly to f (x0). Since
9 is topologically continuous at f (x0), the sequence (g (f (x..))) converges strongly
to g(f (x0)) 2 (go f) (x0). But for all n 2 1, we have that g(f (xn)) = (go f) (xn).
Thus, the sequence ((g o f) (xn)) converges strongly to (g o f) (x0). This is true for
any sequence (xn) in D, that converges strongly to x0. By Lemma 5.1, (g o f) is

topologically continuous at x0.

Corollary 5.2 L6.
ND!) C Dav f l5
0,. ThengOf is!

Definition 5.3 Le
uniformly continua

€115!st inR s

 

Lemma 5.2 Let [

continuous on D if

Proof. First assun

be given in R. Tho

we We that

Hence

Ill palllCular‘

103

Corollary 5.2 Let Df,Dg C R and let f : Df -> R and g : D9 -—+ R be such that
f (Df) C D9, f is topologically continuous on D, and g topologically continuous on

Dg. Then g o f is topologically continuous on Df.

Definition 5.3 Let D C R and let f : D —+ R. Then we say that f is topologically
uniformly continuous on D if and only if for all x E D and for all e > 0 in R there

exists6 > 0 inR such that
yeDand ly-xl<5=>lf(y)-f($)l<€-

Lemma 5.2 Let D C R and let f : D —-> R. Then f is topologically uniformly

continuous on D if and only if for all e > 0 in R there exists 6 > 0 in R such that

x,yEDand0<y—x<6=>]f(y)—f(x)|<6.

Proof. First assume that f is topologically uniformly continuous on D, and let 6 > 0

be given in R. Then by Definition 5.3, there exists 6 > 0 in R such that for all x E D,

we have that

yeDand ly-$|<5=>lf(y)-f(iv)|<6-

Hence

x.y€Dand ly—rvl <6=> |f(y)-f(x)| <6;
in particular,

x,yEDand0<y—x<6=>|f(y)—f(x)|<5.
Now assume that for all e > 0 in R there exists 6 > 0 in R such that

x,yEDand0<y—x<6=>|f(y)—f(x)|<6. (5.1)

We show that f is topologically uniformly continuous on D. So let 2 E D and let 6 > 0

in R be given. Let 6 > 0 in R be as in Equation (5.1), and let w E D be such that

;u‘-z[<6. lfz <

 

ifur<z,letr=u

alleDandlora

Hence I is topolog

 

I
Theorem 5.3 Let
“lllonnly continue

lUPOlWlCfllly uni/or"

 

PlOOl. “9 may as
lllnEZfi lEl In :

l \
'lnl converges Stro

mR'SOletc>0

There Wists N E ‘

ROW let m. n > V

‘ C

104

]w—zl <6. Ifz<w,letx=zandy=winEquation(5.1)toget|f(w)—f(z)| <6;
if w < 2, let x = w and y = z in Equation (5.1) to get If (.2) — f (w)| < 6. Hence for

alle Dand for alle> Oin R, there exists6 >OinRsuch that
wEDand |w—z|<6=>|f(w)—f(z)|<e.

Hence f is topologically uniformly continuous on D.

Theorem 5.3 Let a < b be given in R and let f : I (a, b) —> R be topologically
uniformly continuous on I (a, b). Then there exists a unique function g : [a, b] -—+ R,

topologically uniformly continuous on [a, b], such that

glI(a,b) = f

Proof. We may assume that I (a, b) 79 [a, b]. First assume that I (a, b) = (a, b]. For
all n E Z1", let xn = a+d" (b — a). Then xn E I (a, b) for all n 2 1; and the sequence
(33") converges strongly to a. We show that the sequence (f (xn)) converges strongly

in R. So let 6 > 0 be given in R. Then there exists 6 > O in R such that
x,yE (a,b] and Iy—xl < 6=> |f(y)—f(x)| <6.
There exists N E Z + such that
d” (b — a) < 6.
NOW let m, n 2 N be given. Then
Ixm—xnl :2 |d'"—d"|(b—a) $dN(b—a) < 6.

Thus,
lf($m) - f (xn)| < e for all m,n Z N.

Hence the sequenc
respect to the 0rd:

by

We show now t]

given in R. There 1

 

There exists N E Z

l0“ 1“ fly 6 ]a. b‘

HI : a: then

0<y

Since

3? Obtain that

w‘lx

105

Hence the sequence (f (xn)) is strongly Cauchy. Since R is Cauchy complete with
respect to the order topology, (f (xn)) converges strongly in R. Define g : [a, b] —+ ’R,

by
g($)_{f(x) ifxE(a,b].

_ limnhoo f (xn) if x = a

We show now that g is topologically uniformly continuous on [a, b]. Let 6 > 0 be

given in R. There exists 6 > 0 in R such that
e
m,l/em] and Iy—xl <6=>If(y)-f(:r)l <5.
There exists N E Z + such that

dN(b—a)<6and ]f(xN)—g(a)|<

[\DIm

Now let x,y E [a, b] be such that 0 < y — x < 6. Then y E (a, b]. If x E (a, b], then

rum-gov): = |f(y)-f(r)| < 3 <e.

Ifx=a,then
0<y—a=y—x<6andIy-xN|=|y—a—dN(b—a)|.

Since

0<y—a<6and0<d~(b-a)<6,

We obtain that

]y—xNI = max{y—a,d~(b—a)} —min{y-—a,d~(b-—a)}

< max{y—a,dN(b—a)} <6.

and hence

6

lfly) — f(iENll < 2

Thus.

Hence 9 is topologi

lfil 91 1 ]a. bl “
To show that 91 =
continuous at a an

the sequence (91 (1.

8091:9; and hem

Similarly we ca

[Hill] : (ab).

Examme 5.2 Let

 

lie show that f is
M1 ‘* R Such ti

”ll-ll and let 6;

and let y g (031] be

H
<I\y<6, and

lfl

106

Thus,

|g(y) - g(xll = |f(y) - WIN

3 Mo) -f(x~)|+lf(:r~) —g(a)l
< §+§=e

Hence g is topologically uniformly continuous on [a, b] and g|1(a,b) = f.

Let 91 : [a, b] —-> R be topologically uniformly continuous on [a, b] with 91l1(a.b) = f.
To show that g1 = g, it suffices to show that 91 (a) = g (a). Since 91 is topologically

continuous at a and since (xn) converges strongly to a, we obtain by Lemma 5.1 that
the sequence (g1 (xn)) converges strongly to g1 (a). Thus,

91 (a) = 315.10 91 (en) = "1320f (22,.) = 9 (0).
So g1 = g; and hence g is unique.

Similarly we can show that the result is true for the cases I (a,b) = [a, b) and

1(a, b) = (a, b).
Example 5.2 Let f : (0,1] —> R be given by g(x) = l/x.

We show that f is topologically continuous on (0,1]; but there is no function g :
[0, 1] —> R such that g is topologically continuous on [0,1] and 9l(0.1] = f. Let

:13 E (0, 1] and let 6 > O in R be given. Let

6-min x 6332
_ 2’ 2 ’

and let y E (0,1] be such that 0 < Iy — x] < 6. First assume that that y < x. Then

0< x—y<6,and

 

1 1 1 1 6
|f(y)-f($)| = 5—;<$_,-;=m
S 6 =-2—(:<6

107

Now assume that x < y. Then 0 < y—x < 6, and

1 1 1 1 6
lf(y)—f($)l _ x_y<x_x+6 x(x+6)
6 e
_<_ .
< $2_2<€

Hence f is topologically continuous at x for all x E (0, 1], and hence f is topologically

continuous on (0, 1].

Suppose there exists g : [0, 1] ——» R such that g is topologically continuous on [0, 1]
and 9l(0,1] = f. Then, by Lemma 5.1, we have that the sequence (g (d")) converges
strongly to g (0); i.e. the sequence (d‘n) converges strongly to g (0), which contradicts

the fact that (d"") is unbounded. So there can be no such g.

Definition 5.4 Let D C R be open and let f : D —+ R. Then we say that f is
topologically differentiable at x0 E D if and only if there exists a number f’ (x0) E R

such that the function FM,0 : D —> R, given by

3

[LEM—1 if 22 7e mo
F 1.1120 (x) =
f I (130) if 13 = 1‘0
is topologically continuous at x0. If this is the case, we call [7on the derivate function

off at x0 [10], and f’ (x0) the derivative off at x0.

Definition 5.5 Let D C R be open and let f : D —+ R. Then we say that f is
tapologically differentiable on D if and only if f is topologically differentiable at x for

all x E D.
The following lemma follows directly from Definition 5.4.

Lemma 5.3 Let D, f, :50, F14,0 be as in Definition 5.4. Then we have that

f (x) = f (x0) + F1,“ (x) (x — x0) for all x E D.

108

The following lemma is a direct consequence of Definitions 5.4 and 5.1.

Lemma 5.4 Let D C R be open and let f : D —-+ R. Then f is topologically
differentiable at x0 E D if and only if there exists a number f’ (x0) E R such that for

all e > 0 in R there exists 6 > 0 in R such that

f (37) " f(170)

13—210

—f’(.’130) < 6.

xEDandO<|x—x0|<6=>

Theorem 5.4 Let D C R be open and let f : D —-) R be topologically differentiable

at x0 E D. Then f is topologically continuous at 2:0.

Proof. Let c > 0 be given in R, and let 61 = e/ 2. Since f is topologically differen-

tiable at x0, there exists 61 > 0 in R such that

17) - f (130)

— f’ (x0) < 61.
113—1130

xEDandO<|x—x0|<61=>]f(

Let
6— Hfin{51,m,1} iff’(l‘0)#0
min {51,1} if f’($0) =0

Then 6 > 0, and for all x E D satisfying 0 < |x — x0] < 6, we have that

lf($)-f(-’L‘o)| < 61 l$—$0l+lf'($o)ll$—$ol

< 56+ If’ (mo)|6

|/\
NJImNJ

+ =6.

Hence f is topologically continuous at x0.

Corollary 5.3 Let D C R be open and let f : D —+ R be topologically differentiable

on D. Then f is topologically continuous on D.

109

Theorem 5.5 Let D C R be open, let f, g : D —-> R be topologically differentiable
at x0 E D, and let a E R be given. Then (f +ag) and (fo 9) are topologically
differentiable at x0, with derivatives

(f+09)’($0) = f, (330) +019, (330) and

(f'9)'($0) = f' ($o)9($o)+f($0)9' (1130)-

Proof. Let F130 and 01,30 denote the derivate functions of f and g at .730, respectively.
Then FL,B0 and G1,“ are topologically continuous at x0. By Theorem 5.1, we have

that the function F1,“ + 010on : D —+ R, given by

(Fléco + 001.10) (23) = F1,1‘o (:13) + Gal,“ ((1))

z _
x-Io x—Zo

{ILL—L124“ +a-‘A—Lil-i—H0 ifx¢xo

f’ (x0) + ag’ (x0) if x = x0

 

x—xo

’

{ o+ag)(x)-o+ag)(mo) if a, 75 530

f’ (x0) + ag’ (x0) if x = x0

is topologically continuous at x0. Thus, (f + (19) is topologically differentiable at x0,

with derivative

(f + ag)’ (2:0) = f’ (170) + 09’ (x0) -

Now let H : D —+ R be given by

' 2:20. $° if x 75 x0
H (x) = { .
f' (330) 9 (370) + f (330) 9’ (550) if 3 = $0

110

To show that (f - g) is topologically differentiable at x0, we need to show that H is

topologically continuous at x0. Note that

 

flflgL-‘CFI (130)9(130) if 1. 7e 330
H (x) = { °
f, (170)9(170) + f (330) 9' (970) if 93 = $0

{ngHflflW ifn: #270
f' (330)9(170) 4' f (330) 9' (330) if 13 = 1‘0

= Flam (3)9(1‘0) +f($) 01.30 (1‘)-

Hence H = g(xo)F1,1-o + f (11,20. Using Theorem 5.1, we obtain that H is topologically

continuous at x0. Thus, (f - g) is topologically differentiable at x0, with derivative
(f ' 9), ($0) = H(IE0) = f, (170)9(330) + f (930)9’ (170).

Corollary 5.4 Let D C R be open, let f, g : D —+ R be topologically differentiable on
D, and let a E R be given. Then (f + ag) and (f - g) are topologically diflerentiable

on D, with derivatives
(f+ag)'=f’+ag’ and (f°g)'=f’-g+f-g’.

Theorem 5.6 (Chain Rule) Let Df,Dg C R be open, and let f : Df —» R and
g : Dg —e R be such that f (Dy) C D9, f is topologically differentiable at x0 E Df and
g topologically differentiable at f (x0). Then g o f is topologically difl'erentiable at x0,
with derivative

(9 0 f), (130) = 9' (f (550)) f, (170)-

111

Proof. Let F1,“ and Gl.f(a=o) denote the derivate functions of f at x0 and of g at

f (x0), respectively. Let H : D, —+ R be given by

° x:° x0 ifxyéxo
xxo
H(x)=
9’

(f ($ollf' (are) if a: = as. °

Then
91% if x 75 x0
H (x) = {
9' (f (170)) f’ (550) if$ = $0
WW ifxséxo and f(x)7éf(x0)
= 0 if$¢$oandf($)=f($0)
9' (f (500)) f' ($0) if $ = $0
= Gl,f(:ro) (f (33)) F1,:r:o (It) '
Hence

H : (G1,f($o) O f) ' F1.$0'

Since f is topologically continuous at 2:0 and since G 1, f(xo) is topologically continuous
at f (x0), we have by Theorem 5.2 that (Gl.f(xo) o f) is topologically continuous at
x0. Since F1,” is topologically continuous at x0, so is (Glflxo) o f) - Fm,o = H by

Theorem 5.1. Hence (g o f) is topologically differentiable at x0, with derivative
(9 0 f), ($0) = H ($0) = 9' (f (350)) f' ($0)-

Corollary 5.5 Let Df,Dg C R be open, and let f : Df —-) R and g : Dg —* R
be such that f (Df) C D9, f is topologically differentiable on D, and g topologically

diflerentiable on By. Then g o f is topologically difi'erentiable on Df, with derivative

(90f)' = (g’Of)-f’.

112

Theorem 5.7 Let D C R be open, and let f : D —+ R be such that f is topologically

differentiable and has a local maximum at x0 E D. Then
f ’(1130) = 0.

Proof. Suppose not; then I f’ (xo)| > 0. Since D is open and since f is topologically

differentiable at x0, there exists 6 > 0 in R such that (x0 — 6, x0 + 6) C D and

f(xl - f($0)

:3-130

 

_ f’(g;0) < df’(x0) for all x at x0 in ($0 — 5,370 'l' 5);

which entails that

f(xl — f($0)

III—£130

 

z f’(x0) for all x 34 x0 in ($0 — 5,1130 'l' 5)-

In particular, (f (x) — f (xo)) / (x — x0) has the same sign (that of f’ (x0)) for all x 79 x0

in (x0 — 6, x0 +6); which contradicts the fact that f has a local maximum at x0. Thus,

f’($0) == 0.

Corollary 5.6 Let D C R be open, and let f : D —-> R be such that f is topologically

differentiable and has a local minimum at x0 E D. Then
f’($0) = 0.

Proof. Let g = — f. Then g is topologically differentiable and has a local maximum
at x0. By Theorem 5.7, we obtain that g’ (x0) = 0. Using Theorem 5.5, we finally

obtain that

f’($o) = -g’(xo) = 0-

The following examples show that, contrary to the real case, topological continuity
or even topological differentiability of a function on a closed interval of R are not
always sufficient for the function to assume all intermediate values, a maximum, a

minimum, or a unique primitive function on the interval.

:1

113

Example 5.3 Let f : [0,1] ——i R be given by

_ 1ifx~1
flail‘io ifOSx<<1'

Then f is topologically continuous on [0, 1] and topologically differentiable on (O, 1),

with derivative f’ (x) = 0 for all x E (0, 1). We have that
f(0)=0<d<1=f(1); but f(x)7£dforalle [0,1].

So f does not satisfy the intermediate value theorem on [0, 1]. Moreover, although

f’ (x) = 0 for all x E (0,1), f is not constant on [0,1].

Example 5.4 Let t > 0 be given in Q, and let f : [—1, 1] —» R be given by

_ d‘exp(x)+sign(x)exp(—1/x2) ifx~1
f(x)— d‘exp(x) ifOSx<<1’

where
. 1 i x>0
3’9"”) ={ —1 ifoO'

Then f is topologically continuous on [—1, 1] and topologically differentiable on

(-1, 1) with derivative

f'( ) _ d‘exp(x) + fgsign (x) exp (—1/x2) if x ~1
x d‘exp(x) if0$x<<1

d‘exp (x) + figexp (—1/x2) if x ~1
d‘exp(x) ifOSx<<1
> Oforalle (—1,1).
Moreover, f is strictly increasing on [—-1, 1]. We have that

f (—l) = d'exp(—1)— exp(—1)< d < d‘exp(1)+ exp(—1)= f (1);

but f(x) 75 d for all x E [—1,1].

rV

114

Example 5.5 Let f : [—1, 1] —» R be given by

f(x)=x—§R(x).

Then f is topologically continuous on [— 1, 1]. However, f assumes neither a maximum
nor a minimum on [—1, 1]. The set f ([—1, 1]) is bounded above by any positive real
number and below by any negative real number; but it has neither a least upper

bound nor a greatest lower bound.

Example 5.6 Let f : [—1, 1] —» R be given by

f (x) = Z xud3q" when x = if? (x) + Z xudq".

12:1 11:1

Then f is topologically continuous on [—1, 1] and topologically differentiable on

(—1,1), with derivative
f’ (x) = 0 for all x E (—1,1).

f has neither a maximum nor a minimum on [—1, 1]. Moreover, f is not constant on

[—1, 1] even though f’ (x) = 0 for all x E (—1,1).

Example 5.7 Let f,g : [—1, 1] —» R be given by

f (x) = x and g (x) = x + d3"($)+1.

Then f and g are both topologically continuous on [-1, 1] and topologically differen-

tiable on (—1,1), with derivatives

f’ (x) = 1 = g’ (x) for all x E (—1,1).

 

115

So f and g are two primitive functions of 1 on [-—1, 1] that do not differ by a constant.

In the following section, we introduce stronger smoothness criteria on R and use

them to try and extend the common theorems of real calculus to R.

5.2 Continuity and Differentiability

Definition 5.6 Let a < b be given in R and let f : I (a, b) —+ R. Then we say that f

is continuous on I (a, b) if and only if there exists M E R, called a Lipschitz constant

off on I (a, 6), such that [10]

f(y)-f(x)
y—x

 

SMforallxyéyinI(a,b).

 

Lemma 5.5 Let a < b be given in R and let f : I (a, b) —+ R be continuous on

I (a, b). Then f is topologically uniformly continuous on I (a, b).

Proof. Let M be a Lipschitz constant of f on I (a, b), and let 6 > 0 be given in R.

Let 6 = e/M. Then 6 > 0, and for all x,y E D satisfying 0 < y — x < 6, we have that

lf(y)-f($)lSM(y—x)<M6=e.

Hence, using Lemma 5.2, we obtain that f is t0pologically uniformly continuous on

D.

Lemma 5.6 Let a < b be given in R and let f : I (a, b) —> R be continuous on

I (a, 6). Then f is bounded on I (a, b).

Proof. Since f is continuous on I (a, b), there exists M E R such that

]f(y)-f($)
31

—$

 

SMforallxaéyinI(a,b).

 

116

 

 

 

 

 

 

Thus,
x - 9:2 —
lf() {.92} ngoranxsb ammo);
x——2- 2
andhence
]f(x)]£]f(b;a) +Mx-b‘a' < f(bga) +Mlb—al

 

 

 

 

 

for all x E I(a,b).

Lemma 5.7 Let a < b be given in R and let f : I (a, b) —-» R be continuous on I (a, b)
with Lipschitz constant M. Let x E I (a, b) be given, let r E Q, and let h E R be such
that ]hl << (1’ and x + h E I(a,b). Then

I (17 'l' h) =r+A(M) f (~73) -

Proof. If h = 0, we are done. So we may assume that h 74 0. Thus,

|f(:c+h)-f(a:)
h

 

3M; andhence |f(x+h)—f(x)| SMlhI.

 

Thus,
/\(f($+h)-f($)) ZMMlhI) =>\(M)+A(h) >>‘(M)+r.

which entails that f (x + h) =,+,\(M) f (x).

Lemma 5.8 (Remainder Formula 0) Let a < b in R and let f : I (a, b) ——> R be
continuous on I (a, b) with Lipschitz constant M. Then for all x, y E I (a, b), we have

that

f(y) = f(x) +ro(x.y)(y-rr). with A(ro(x,y)) 2 MM)-

Proof. Let x,y E I (a, b) be given. Let

nettle) if”,

= 31-3 .
T0($,y) {0 lfyz—‘IB

 

Then fl?!) =.

have that

Theorem 5.8

I(a.b), and le-

Proof. Since
f (y) -
y -
Let

HellCef+ Og

bOllllded 01] I

117

Then f (y) = f (x) + r0 (x, y) (y - x). Moreover, since f is continuous on I (a, b), we

have that

ho (x,y)l S M; and hence A(ro (x, y)) 2 /\(M).

Theorem 5.8 Let a < b be given in R, let f,g : I (a, b) ——> R be continuous on

I (a, b), and let a E R. Then f + (19 and f - g are continuous on I (a, b).

Proof. Since f and g are continuous on I (a, b), there exist M1, M2 E R such that

]flyl-fl“) SMgforallx7éyinI(a,b).
y—x 9'3

 

SMI and ]g(y)-g($)

 

Let
M = max {M1, M2}.

Then

 

]f(y)-f(x)
y—x

 

S Maud Infill-g(x)

< ' .
y—x _Mforallx75y1n1(a,b)

 

Now let x 7E y in I (a, b) be given. Then

] (f + as) (y) - (f + as) (a?)

 

]f(y)+ag(y) -f($) -ag(-'L')

 

 

 

 

 

y-x y—x
S f(y)-f(x) +I04]g(y)-g($)
y—x y—x
g fl+kmhf

Hence f + ag is continuous on I (a, b) with Lipschitz constant (1 + Ial) M.

Since f and g are continuous on I (a, b), we have by Lemma 5.6 that f and g are

bounded on I (a, 6). Hence there exists Mo E R such that

|f(x)| S M0 and Ig($)l S M0 for alleI(a,b).

118

Now for all x 7Q y in I (a, b), we have that

Us) (.21) - (f-g) (Iv)

 

 

]f(y)9(y)-f($)9($)

 

 

 

 

 

y—x y—x
= ]f(y)(g(y)-g(m))+(f(y)-f(x))g(x)

y—x
S ”(w|]9(y):9( $)+|] 9($)| f(yz]:£(x)

 

 

S M0M2 + MoMl

S 2M0M.

Hence f . g is continuous on I (a, b) with Lipschitz constant 2M0M.

Theorem 5.9 Let a < b and c < e in R be given, and let f : [1(a, b) —+ R and
g : 12(c, e) —i R be such that f(11(a,b)) C 12(c, e), f is continuous on [1(a, b) and g

continuous on 12(c, e). Then g o f is continuous on Il(a, 6).

Proof. Let M f and Mg be Lipschitz constants of f on 11 (a, b) and of g on 12(c, e),

respectively. Let x 75 y be given in [1(a, b). First assume that f (y) = f (x). Then

](QOf)(?/) - (9°f)(x) = ]g(f(y)) -g(f($))
y-x y-x

 

= 0 s Mng.

 

 

Now assume that f (y) sé f (x). Then

](QOny) - (9°f)($) = ]g(f(y)) -g(f($))
y—x y—x

 

 

 

 

g (f (31)) - g (f (x)) f (y) - f (x)
f(y)-f(x) y-x

g(f f(y))-g(f) (13)) ]f(y)-$ f(x)
f(y)-
< Mng.

 

 

 

 

 

 

Thus, for all y

and hence (g C

Theorem 5.1
I(a,b). Then

that

Proof. We in

which exists b

on iMl by Le

ROW assume .

0‘;

119

Thus, for all y 75 x in [1(a, b), we have that

(90f)(y)-(9°f)($)
y—x

 

S Mng;

and hence (g o f) is continuous on 11 (a, b), with Lipschitz constant Mng.

Theorem 5.10 Let a < b be given in R and let f : I (a, b) —+ R be continuous on
I (a, b). Then there exists a unique function g : [a, b] —+ R, continuous on [a, b], such

that

gll(avb) = f'

Proof. We may assume that I (a,b) 7é [a, b]. First assume that I (a, b) = (a, 6]. Let
f0 = mirawtb— a»,

which exists by the proof of Theorem 5.3 since f is topologically uniformly continuous

on (a, b] by Lemma 5.5. Define g : [a,b] —> R by

g(x)={ f(x) ifxE(a,b].

f0 ifx=a

It remains to show that g is continuous on [a, 6]. Let M be a Lipschitz constant of f
on (a, b] and let x 79 y in [a, b] be given. Without loss of generality, we may assume

that x < y. Assume that a < x, then x, y E (a, b], and hence

]g(y) -g(-'B) f(y) -f(-'r)

 

 

SMS2M.
y—x y—x

 

 

Now assume that x = a. There exists N E Z + such that
dN(b—a) <y—aand ]f(a+d~(b—a)) —g(a)] S M(y—a).

Then it follows that

0<y—(a+dN(b—a)) <y—a;

and hence

9(y)-.
y—.

Thus. for all

Hence 9 is C(

Similarly.

and1(a,b) :

Defiflltlm] E
I(a,b). The
“function fr

IEI(alb):l

120

 

 

and hence
]g(y)-g($) ___ f(y)-9(a)
y—x y—a

 

 

f(y)—f(a+d”(b-a))+f(a+d”(b-a)) —g<a)

 

 

 

 

 

 

 

 

 

 

y—a
S ]f(y)-f(a+d”(b-a)) +]f(a+d”(b-a))-g(a)
y—a y—a
< f(y)-f(a+d”(b-a)) +M
_ y—(a+dN(b-a))
S M+M=2M.
Thus, for all x 75 y in [a, b], we have that
]g(y)-g(x) <2M.
y-x ‘

 

Hence 9 is continuous on [a, b] with Lipschitz constant 2M, and g|(a,b] = f.

Similarly, we can show that the result is true for the cases when I (a, b) = [a, b)

and I(a, b) = (a, b).

Definition 5.7 Let a < b be given in R and let f : I (a, b) -—+ R be continuous on
I (a, b). Then we say that f is difierentiable on I (a, b) if and only if there exists
a function f’ : I (a, b) -—» R, called the derivative of f on I (a, b), such that for all
x E I (a, b), the derivate function FL; : I (a, b) —+ R [10], given by

y—l‘

FL: (:9) =

7

{Mt-“9 ifyséx
f’($) ifs/=17

is continuous on I (a, 6).

Lemma 5.9 Let a < b be given in R and let f : I (a, b) -> R be differentiable on

I (a, b). Then f is topologically differentiable on (a, b).

121

Proof. Let x E (a, b) be given. Then the derivate function FM. is continuous on
I (a, b). By Lemma 5.5, F1; is topologically continuous at x. Hence f is topologically
differentiable at x. This is true for all x E (a, 6); hence f is topologically differentiable
on (a, b).

The following theorem is a generalization of a similar result in [5] and is a central
theorem because it reduces computing derivatives to mere arithemtic operations and

thus allows rigorous study of differentiation [38, 39, 40, 42].

Theorem 5.11 (Derivatives are Differential Quotients) Let a < b be given in
R and let f : I (a, b) —r R be differentiable on I (a, b). Let x E I (a, b) be given, let
F1; be the derivate function of f at x, and let M1,; be a Lipschitz constant of Fm.

Let r E Q be given and let h E R be such that O < |h| << d' and x+h E I(a,b). Then

f’ (...) mm...) f (I + h}, " f (“3);

 

which means that

f(x+h)-f(x)
h

 

— f’ (x) << Mmd'.

Proof. Since F1; is continuous on I (a, b), we have using Lemma 5.7 that

1713(2)) =r+A(M1.,) F1,x (II: + h),

where

 

FL; (x) = fI (x) and F13 (x + h) = f (x + h; — f (x).

This finishes the proof of the theorem.

Theorem 5.12 (Remainder Formula 1) Let a < b be given in R and let f :
I (a,b) -e R be differentiable on I (a, b). Let x E I (a, b) be given, let F1; be the

 

derivate font

ye I (ab). 1
Ho) = f
Proof. Since

F14

Thus.

and hence

Theorem 5

[(0.6), and

derivatives

PIOOf. Let
Off and g a]

122

derivate function of f at x, and let M1,; be a Lipschitz constant of F13. Then for all

y E I (a,b), we have that
f (y) = f (x) + f’ (x) (y — x) + r1(x.y)(y — 2:)". with A(r1(x,y)) 2 A (ll/11.x)-

Proof. Since F14.- is continuous on I (a, b), we have by Lemma 5.8 that
171.1(9) = Fm (1‘) + 1"1 (33,?!) (y — 1‘), With A(7‘1($,y))2 A (Mm)-
Thus,

f(y)-f(x)

y-III

 

= f’ (It) +T1 (x,y) (y - 3'3);
and hence

f(y) =f($)+f’($)(y-$)+r1($,y)(y-$)2-

Theorem 5.13 Let a < b be given in R, let f, g : I (a, b) —+ R be differentiable on
I (a, b), and let 0: E R. Then f + cry and f - g are differentiable on I (a, b), with

derivatives

(f+ag)'=f’+ag’ and (f-g)'=f’°g+f-g’.

Proof. Let x E I (a, b) be given, and let FL: and 01.: denote the derivate functions
of f and g at x, respectively. Then F1; and GM are continuous on I (a, b). It follows

by Theorem 5.8 that the function F1; + 001,3 : I (a, b) —> R, given by
(FL: + 001.2) (y) = F1; (31) + 0101.2- (y)
+a — x -—a x -
{ y_z 1f y 75 x

f’ (a?) + 09’ (fit) if y = 16

 

y—IE

)

{ sworn—mama ify 7g ,3

(f’ + 019’) (93) if y = :13

 

is continuous

on I (a. b) wit

Now let I

We show tha‘

Hence

Since I, F,

We for an 1

Theorem 5
my ‘ 12h:

““19 d136,,

123

is continuous on I (a, b). This is true for all x E I (a, b). Hence f +ag is differentiable

on I (a, b) with derivative (f + ag)’ = f’ + ag’.

Now let x E I (a, b) be given, and let H3 : D ——> R be given by

MW ifyséx
Hx(y)={ .
f’(x)g(x)+f(x)g’(x) ify=x

We show that H1. is continuous on I (a, b) for all x E I (a, b). Note that

MW ifyséx
Hz(3/) = {
f'($)9($)+f(:c)g’(a:) ify=x

y—SB

{Warmnmm ifs/#2:
f'($)9($)+f($)g'(x) ify=x

= F1,.(y)g(-r) + f (31) 01,1: (31)-

Hence

Ha: = 9($)F1,z + f ' 01,:-

Since f, FL, and G1,; are continuous on I (a, b), so is Ht by Theorem 5.8. This is

true for all x E I (a, b), and hence (f - g) is differentiable on I (a, b), with derivative
(f - 9)’ (II?) = f' ($)g (93) + f (x) 9' (1‘) for 311$ E I (a,b)~

Theorem 5.14 (Chain Rule) Let a < b and c < e in R, and let f : 11(a, b) —-+ R
and g : 12(c, e) —-> R be such that f (11 (a, b)) C [2(c,e), f is differentiable on Il(a, b)

and g difi'erentiable on 12(c, e). Then go f is differentiable on II (a, b), with derivative

(90f)' = (g'Of)-f'.

LetIEI

Then

The fol]

hinctions an d

0“

Efrem 5.23

124

Let x E I (a, b) be given, and let H1. : D -—r R be given by

° ,:,° 3 ifs/#2:
Hx(y)=
g]

(f (x))f’ (as) ifx = as. '

Then

y-I

{WM ifs/79$
g’(f($))f’($) ifs/=25

“WHEHJ ify¢xandf<y>¢f<x>
= 0 ify¢xandf<y>=f<x>
g' (f (x))f’ (x) if y = x

= Gm.) (f (31)) FL: (1)).

where FL; is the derivate function of f at x, and G1.f(x) the derivate function of g at
f (x). Hence

H. = (0W) 0 f) -F.,..
Since f is continuous on I 1 (a, b) and since G1.f(:c) is continuous on 12(c, e), we have by
Theorem 5.9 that (Gum o f) is continuous on [1 (a, b). Since Fm is continuous on
11(a,b), so is (G1, f(x) 0 f) -F1,x = H, by Theorem 5.8. Hence (g o f) is differentiable

on Il(a, b), with derivative

(90f)'($) = HA1?) =9’(f($))f’($)

= ((g' 0 f) - f’) (x) for all x E Il(a, b).

The following result provides a useful tool for checking the differentiability of
functions and will be used frequently later, as in the proofs of Theorem 5.20 and
Theorem 5.23 and in Example 5.10.

Theorem
[(0. b). So,
that

flmfu

Proof. \V

given by

is continuo

SOletl-E

CUHSldered‘

ASafir

125

Theorem 5.15 Let a < b be given in R and let f : I (a, b) —) R be continuous on
I (a, b). Suppose there exists M E R and there exists a function g : I (a, b) —> R such

that

 

f(y3):£($) —g(x) gM|y—x| forallyaéxin1(a,b).

Then f is differentiable on I (a, b), with derivative f’ = 9.

Proof. We need to show that for all x E I (a, b), the function F14: : I (a, b) —> R,

given by

y—JZ

F1; (.7!) =

,

{Lama ify7éx
g(x) ifs/=1:

is continuous on I (a, b). It is sufficient to show that for all x E I (a, b), we have that

Fl,x (y) —' Fl,x (Z)
y—z

 

S d"1M for all y 75 z in I(a,b).

 

So let x E I (a, b) be given; and let y 34$ 2 be given in I (a, b). Four cases are to

considered.

As a first case, assume that y = x. Then

 

 

 

 

 

 

 

 

 

 

F1. 0) — F... (2) = F1.x($) — F1... (z) _ 9 (x) — Late
y—z x—z x—z
M — g on]
lz - ml
_ M = M
lz - ml
3 d'lM.
As a second case, assume that z = x. Then
F1,3(y) - F1.z(Z) F1,m(y) - F1... (22) My; 3 - 9 (fr)

 

 

 

 

 

 

 

 

y-z y—x y-x

As a thir
]y-I]. The

then ]y— I]

if); (if)

Hence

FlIlaHy9 a:

1,;-

126

 

_ |M,:.x -g<x)|
Iy-xl
_ W=M
ly-xl
g d’lM.

As a third case, assume that y sé x # z and Iy — 2| is not infinitely smaller than
Iy — x]. Then Iy — z] is not infinitely smaller than ]z —— x]; for if Iy — 2] << |z - x],

then Iy — x] = ly — z + (z — x)| z Iz — x] > Iy — z], a contradiction. Thus,

 

 

 

 

 

 

 

 

|F1,x(y)—F1.z(2)l z region) _f(z::£(x)
_ f(y)-f($)_ x _ f(Z)-f(x)_
— ( y”, g()) ( z-.. g(x))l
f(y)-f(x) f(z)-f(x)
g y_$ —g@)+] z—x *9W4

 

 

< M]y-x]+M|z—x|
S d”1M|y-—z|sinced'lly—z]>>|y—x|+|z—x|.

Hence

Fl,x (y) - Fl,x (Z)
y—Z

 

g d‘lM.

 

Finally, assume that yyé x # z and Iy—zl << Iy—xl. Then
z—x=z—y+(y—x)zy—x.
Thus

lF1,x (y) _ Fl,x (Z)l

 

]f(y)-f($)_f(Z)-f($)

y—x z—x

 

 

= ]f(y)-f(x)_f(z)-f(x)y-w

y—x y—x z—x

 

By hypothes:

there

Thus,

Since ]2 C, I]

lll ‘ 33]] We 01

m“ finisl

1981

127

.mm—fai,Ha—fooo+g;g)

y—x y z—x

 

f(y)-f(2)_f(Z)-f($)y-z

y—x y-x z—x

 

 

 

f(y)-f(Z)y-z_f(Z)-f(x)y-z

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

y—z y—x z—x y—x
: y-z f(y)-f(2)_f(Z)-f($)
y—x y—z z—x '
By hypothesis, we have that
f(y)-f(2)_ f(z)-f($)_ ,
y—z —g(z)+r1and z-x —g(z)+r2,
where
|r1|SM|y—z| and |r2|SM|z—x|.
Thus,
-2
inter—asou =l:_$ln—rn
-z
s :_$<un+um

y—z
S M(ly-z|+lz-$I)-

 

 

 

Since [2 — x] z |y — x|, we obtain that |z — x] < d‘1|y — x] /2. Also, since |y — z] <

|y — x], we obtain that ]y — z] < Iy — x] < d’1 Iy — $I /2- Therefore,

y—Z
y—JI

Md"1|y - 39| = d‘lM Iv - ZI;

 

Wu (3!) - F1,z(Z)l S

 

 

and hence

F1,x (y) _ F1,x (Z)
y—z

 

g d'lM.

 

 

This finishes the proof of the theorem.

128

Remark 5.1 The proof of Theorem 5.15 is yet another example of how the non-
Archimedean properties of R allow us to obtain results that would not hold in R or in
any other Archimedean structure. In the third and fourth cases above, the existence of
an infinitely large number (d‘1 in our proof) was essential to get the final inequality.
Indeed we could replace d"1 in the proof of Theorem 5.15 by any positive infinitely

large number without having to change anything else in the proof.

5.3 n-times Differentiability

Definition 5.8 Let a < b be given in R, and let f : I (a, b) —-> R. Let n 2 2
be given in Z+. Then we define n-times differentiability of f on I (a, b) inductively
as follows: Having defined (n — 1)-times difierentiability, we say that f is n-times
differentiable on I (a, b) if and only if f is (n — 1)-times difi'erentiable on I (a, b) and
for all x E I (a, b), the (n — l)st derivate function F _1,1 is diflerentiable on I (a, b).

For all x E I (a, b), the number
f(n) (33) = "Wei—1,2: (33)

will be called the nth derivative of f at x and the derivate function Fm, of F -1“, at

x will be called the nth derivate function of f at x [10].

In connection with the derivate functions, we introduce the secants of different

orders.

Definition 5.9 Let a < b be given in R and let f : I (a, b) ——> R. Then for all
x E I(a, b), the function Sm : I(a, b) \ {x} -—> R, given by

51,2:(9) = “31;: g(x),

 

will be called the first secant of f at x.

 

Definition 5.
be n-times dz]
denote the firs
define Sr; : I (
rolled the lth .

lemma 5.10
be n-times dz“
tive functions

first.....nth d

129

Definition 5.10 Let a < b be given in R, let n E Z+ be given and let f : I (a, b) —e R
be n-times difi‘erentiable on I (a, b). Let x E I (a, b) be given, and let F1,,,,...,F,,,z
denote the first,. . ., the nth derivate functions off at x. For alll E {2, . . . ,n + 1},
define SL3 : I (a, b) \ {x} —+ R to be the first secant of F1-” at x. Then 51.2: will be
called the lth secant of f at x.

Lemma 5.10 Let n E Z+ be given, let a < b be given in R, and let f : I (a, b) —r R
be n-times differentiable on I (a, b). Let f’ ,..., f("l denote the first,. . .,nth deriva-
tive functions of f on I (a, b), and for all x E I (a, b), let F1,x,...,F,,,,, denote the

first,. . .,nth derivate functions of f at x. Then for all x,y E I (a, b), we have [10] that

f®)==f@)+fls@Hy-fl

 

Proof. By induction on n. The assertion is true for n = 1. Suppose it is true for n = l
and show it is true for n = l + 1. So let f be (l + 1)-times differentiable on I (a, b).

Since f is l-times differentiable on I (a, b), we have by the induction hypothesis that
lfij)!(x . r
f (y)= fx() + Z— x—-)’ + FL; (y) (y — x) for all x, y E I (a, b) . (5.2)

Since f is (l + 1)-times differentiable on I (a, b), we have that Fm is differentiable
on I (a, b), with derivative Fit (x) = fa“) (x) / (l+1)! and with derivate function

Fm”, the (l + 1)st derivate function of f at x. Thus,

Es@)==E;@)+EHs@Hy-fl

= {1,z@)+EHsWHy-fl

(ox
= f l_._!__( )+E+m(y)(y— x) forallx,,yEI(a b) (5-3)

130
Substituting Equation (5.3) into Equation (5.2) yields
1 f0) a;
My +2 j,( )(y
j=1

So the assertion is true for n = l + 1; and hence it is true for all n E Z + .

 

—x)j + F1“; (y) (y — x)”1 for all x, y E I (a, b).

Corollary 5.7 Let a < b be given in R, let n E Z+ be given and let f : I (a, b) —r R
be n-times differentiable on I (a, b). Then for all I E {2, . . . ,n + 1} and for all y 75 x
in I (a,b), we have that

f(y)- ' ”3121(— .3):-

Sl,x (y) = E;0_J 1'),

 

Proof. Let y # x in I (a, b) and l E {2, . . . ,n + 1} be given. Then, using Lemma
5.10, we obtain that

Fl~l,x(y) " FI—1,z($)
y - x

_ 1 f(y)-2‘1},(’}’(y-$)j_f(“"(x)
_ y-m (tr-=13)l1 (l-l)!

31,2: (I!)

 

f(y)- ‘ 194% -..»):-

j=0

 

(y - x)’
Corollary 5.8 (Remainder Formula n) Let a < b be given in R and let f :
I (a, b) —> R be n-times differentiable on I (a, b). Let x E I (a, b) be given, let F",I
be the nth order derivate function of f at x, and let Mm, be a Lipschitz constant of
sz. Then for all y E I (a, b), we have that

 

n 0') .
f (y) = f GEM;1 f 3.53:) (y - x)’+rn (w. y) (y - :13)"+1 . with A (cu (m, y)) 2 A (Mm) -

Proof. If y = x, there is nothing to prove; so we may assume that y 75 x. Since F,”

is continuous on I (a, b), we have by Lemma 5.8 that

Fun: (31): Fm: (x) + rn (x, y) (31— 17), With A (Tn ($131)) 2 A (Mme) -

Using Lemn

Also. from I

Thus,

"h

and hence

Theorem 5
R be n.-tz'me
Tamil-tel];

.
I
I

”Till d67‘tuat2‘

131

Using Lemma 5.10, we have that

f(y)-f(x)- "-lnfl—Uty y—xrfi

 

 

 

Fri :1: = j=1
Also, from Definition 5.8, we obtain that
, f ("l (iv)
an’lix (III )2 Fn— 1, x (:1: ): n! '
Thus,
(1')
f (y) - f ($)- 2": L961 - ac) _f(") ___—(_x)
+ rn x — x ;
(y_ x). ( y) (y )

and hence

 

n 0) x . 1
f j,( lty—x>’+r.<x.y)<y—x>"+ .

f (y) = f ($) + Z

j=1

Theorem 5.16 Let n E 2”" be given; let a < b be given in R; let f,g : I(a, b) —+
R be n-times differentiable on I (a,b), with derivatives f’,. ., f(") and g’,. g,(")
respectively; and let or E R be given. Then f +09 is n-times diflerentiable on I (a, b),

with derivatives

(f+ag)(l) = f(z) +019“) for alll E {1,...,n}.

Proof. By induction on n. The assertion is true for n = 1 by Theorem 5.13. Suppose
it is true for n = m and show it is true for n = m + 1. So we have that f and g are
(m + 1)-times differentiable on I (a, b). By the induction hypothesis, we have that

(f + 09) is m—times equidifferentiable on I (a, b) with derivatives
(f + 019)“) = f(z) + Org“) for all I E {1, . . . ,m}. (5.4)

Now let x E I (a, b) be given. Since f and g are (m + 1)-times differentiable on

I (a, b), we have that the (m+ 1)st derivate functions of f and g at x, Fm+1,z,Gm+1,z :

132

I (a, b) —-r R, given by

— x— m (0’ -.'Bl
f(y) f() 215.56% > ifygcéx

 

 

(v-x)"‘“

Fm+1,x (y) =

(n+1) I .

(m+1)! If y = x

and
_ _ ... (0 z _x ,
g(y) 9(1) (335'; inf-1(2) ) if y 5’4 x

Gm+1,$ (y) = i

(m+l) :1: .

(m+1)! If y = ID

are continuous on I (a, b). By Theorem 5.8, we have that Fm+1,x+osz+1,x : I (a, b) —>

R, given by

(Fm-+1.2: + aGm+l.2) (y) = Fm+1,x (y) 'l' aGm-l-LI (y)

f(y)+ag(y)—(f(z)+ag(z))-Zi:_1 WW” if y 75 a;

(rr--'I=)"“C1

(m+1)z (m+”; .
I (3.12%. U lei/=2:

_ a :2: - m a (I) a: _II
(f+09)(y) (f‘l‘ 9)( ) 21—1iLt—gll—Llw ) ify 75 a;

(ii-z)”

f(m+1)Lg;C:%(!m+1)(¢) if y = a:

is continuous on I (a, b), where use has been made of Equation (5.4). Hence f + (19

is (m + 1)-times differentiable on I (a, b), with (m + 1)st derivative

(f + ag)(m+1> (x) = f<m+1> (x) + ag<m+1> (x) for all x e I (a, b).
So the assertion is true for n = m + 1, and hence it is true for all n E Z +.
Definition 5.11 Let a < b be given in R, and let f : I (a, b) -—> R. Then we say

that f is infinitely often differentiable on I (a, b) if and only if for all n E Z“, f is

n-times difi‘erentiable on I (a, b).

133

Using Theorem 4.11, we obtain the following result.

Theorem 5.17 Let a < b be given in R, and let f : I (a, b) —+ R be infinitely often

differentiable on I (a, b). Let x0 E I (a, b) be given, and let

A0 = lim sup (—l\ (f(") ($0») .

n—roo n

 

Then 23,10 f("l (x0) / (n!) (x — x0)" converges strongly if A (x —— x0) > A0, and it is

strongly divergent if A (x — x0) < A0.
Using Theorem 4.12, we obtain:

Theorem 5.18 Let a < b be given in R, and let f : I (a, b) —+ R be infinitely often

differentiable on I (a, b). Let me E I (a, b) be given, and let

A0 = lim sup (-A (f(") (330)) ) .

 

n—+oo n

Let x E I (a, b) be such that A (x — x0) = A0. For all n _>_ 0, let

b = dnAo f(n) ($0) .

n!
Suppose that the sequence (b,.) is regular; and write U;’,°=Osupp(bn) = {q1, q2, . . .} with

(In < are if 2'1 < 1'2. For all n 2 0, write bn = Z3111 bndej where bu,- = b.. [q,-]; and let

T _ 1
sup {limsupndoo |bnj|1/” :j 2 1}.

 

Then 23;], f(") (x0) / (n!) (x — x0)" converges weakly if ](x — x0) [A0]] < r, and it is

weakly divergent if ](x - x0) [A0]] > r.

The MO
conierges, tl
mainder tern

a criterion iC

Example 5.

flmfsm:

llius.

Hence the in

E"ample 5.

lhen f is in!

First “’9 shor

r (1

134

The following two examples show that, even when 22:0 f ("l (x0) / (n!) (x — x0)"
converges, the series need not converge to f (x). It converges to f (x) only if the re-
mainder term rn (x0, x) (x — x0)"+1 converges to O; and Theorem 5.19 below provides
a criterion for that.

Example 5.8 Let f : [—1, 1] —-> R be given by

f(x) ___ { exp(—1/x2) ifx~1
O if0S|x|<<1'

Then f is infinitely often differentiable on [—1, 1], and we have that
fl") (0) = 0 for all n 2 1.

Thus,

 

f: f (”l (0)

n' x" converges strongly to 0 for all x E [-1, 1] .

n=0

Hence the limit is equal to f (x) if 0 S |x| << 1 and is different from f (x) if x ~ 1.

Example 5.9 Let f : [0,1] —> R be given by

0 fix=0
on

yzli-Ji—fj ifn_>_1 andn-lSA(x)<n

Then f is infinitely often differentiable on [0, 1] with derivatives at 0 given by
f“) (0) = d“ for all 1 2 1.
First we show that f is differentiable on [0, 1] with derivative

(1‘1 ifx=0

Ey=1%ifn21mdn-1SA(x)<n

135

We will show that

f(y)-f($)

y—x <d‘11/4Iy—xl forallxyéyin [0,1],

 

— 91 (17)
So let x 7E y in [0, 1] be given. Five cases are to be considered.

First case: x = 0 < y. In this case, f (x) = f (O) = 0 and 91 (x) = gl (0) = d‘l; so

 

 

 

f(y)-f($)_gl($) ___ |£_(_l_l_l__d—1
31—37 y
0 ifOSA(y)<1
= ggflfi: ifmZ2andm—1SA(y)<m
0 ifOSA(y)<1

3:20;}: ifmZ2andm—1SA(y)<m
Note that for all m 2 2 and for m— 1 S A(y) < m, we have that

A(d"jyj‘2) -_- —2+ (j — 2),\ (tr-1y) _>_ —2 for allj e {2,...,m}.

Thus,

 

m -J' 1-2
A(Zd y )2—2forallm22;

,1. j!

and hence

< d—11/4y = 61—11/4 ly _ III] .

 

|f<v;:;<x>-..s

 

Second case: 0 = y < x. In this case, we have that

 

 

 

 

 

“2:53) we) =|ff)—g.(x)
0 if0SA(x)<1

 

by?“ —Zy=2dT’jf—j)',—‘ ifnZ2andn—1 SA(x)<n
0 ifOSA(x)<1

ELK—031): - i) We” if n 2 2 and n — 1 s A (2:) < n

 

136

Foralan2andforn—1SA(x)<n,wehavethat

A (d’jxj_2) = —2+ (j — 2)A(d‘1x)_>_ —2 for all j e {2,...,n}.

 

Thus,
" 1 1 _. ._2
A(Z(fl—F)d JIL‘J )_>_—2foralln22;
j___2 . .
and hence
f (y; : i (x) - 91(22) < it‘ll/4a: = d—11/4 ly — xl.

 

Thirdcase: x>0,y>0, andn—l S A(x),A(y) <n;forn2 1. Inthiscase, we

have that

 

 

0 ifn=1
= ]{ “(d—11M)” 7! iii: ifn22l

J=2 (j—l)!

 

 

o . __ un=1
' 0‘11" 2'! 0'; (IL—‘2’ —ja:J'-1) ifn 2 2 '

y—x
So it remains to show that

11/4 11 —j '_ j .
d Z:d (y’ x _ij_1)

y—xj=2 j! 31—3

 

 

<1foralln22.

 

 

We have that

yr-xr
y—x

 

— ix“ = 111—1+ 11343: + - . - + yxj"2 + xj_1— jxj‘l
= 21"“ + 111% + - - - + yr"? — (j-—1):z:"’1

= (y"1 — xj‘l) + x (yj-2 — xj'2) + - - - + x"—2 (y — x).

Thus, for all j E {2, . . . ,n}, we have that

d11/4d’j yj—xj .._,
A(y—xr (,_. ))

 

 

_ _1_1_j+,((yj'1—x"“)+x(y"‘2—zj"2)+---+arj‘2(y—x))

 

 

 

 

 

 

 

11 . . . . .
= T—j+/\((y'7_2+'”+1‘7_2) +x(y’_3+°~+xJ—3)+-~+x’"2)
_ 3 __1 j-‘2 _1 j—2 _1 j"2
— Z+’\(((d y) +-~+(d x) )+---+(d x) )
2 3 since A(x) Z 1 and A(y) Z 1.
Hence
11/4 n —j '_
A d 2d,. (yJ x —jx"1) ngoralanZ.
y — x]:2 J. y — x 4
Therefore,
11/4 1: -—j '_ j .
d 2d,. (3,] x —jx’"1) <1for alln22;
y-mfigr y-x
and hence
“32:51? —91(:c) < d‘lmly-xl.

 

Similarly, we can show that

]f(y)-f(w)
y—x

< d—11/4ly _ III]

 

— 91 (1‘)

 

for the remaining two cases, namely the case when x > 0, y > 0, and n — 1 S
A(x) <nSm—1SA(y) <m,forn2 1; andthecasewhenx>0,y>0, and
m — 1 S A(y) < m S n — 1 S A(x) < n, for m 2 1. Thus, f is differentiable on
[0,1], with derivative f’ (x) = g1 (x) for all x E [0,1]. In particular, we have that
f4m=gdm=d4-

Similarly, we can show for all I _>_ 2 that f is l-times differentiable on [0, 1], with
l-th derivative

" if x = 0

d
0 ,. ifOSA(x)<l—1
Z?=‘%rfii ifnZlandn—lSA(x)<n

f") (x) =

138

Since

 

rims... (_A (fit) (on) = limsup (1) =1,

we obtain that

 

ll x' converges strongly for O < :1: << d.

°° f") (0)
However, for no such x does Effie f (I) (O) / (l!) xl converge to f (x). This is so because,
as we will show below, for 0 < :1: < d the remainder term r1(0,x)x'+1 does not
converge to 0 as 1 goes to 00. So let x E [0, 1] be such that 0 < x < d. There exists
m 2 2 in Z+ such that m — 1 S A (x) < m. Then for all I > A (x) + 2, we have that

l d-j

 

 

0, = .———r;
7‘1( 33) jzgng-IJM’J-l-l-J
so
_' _ ' _ m 1
T1(0$)$l+1= i (Th: 2‘: (d Ix)3z(d11:) +
j=m+l~7!x_] j=m+1 3' (m+1)!

since 0 < d‘lx << 1. Thus,
A (r; (0,x)x'+1) = (m+ 1) (A(x) — 1) < (m+ 1)A(x) < 00 for all I > 2A(x) +1,

which entails that

(lim r; (O, x) x1+1 7h 0.

As an example, let x = d2; then

°° f(l)(0)$1 _ °° _jds _ °° d‘

' _
l=0 1' l

Theorem 5.19 Let a < b be given in R and let f : I (a, b) —-> R be infinitely often
differentiable on I (a, b). Let x0 E I (a, b) be given and for each 1 E Z J”, let F1,“ denote

the lth derivate function of f at x0 [10]. For each l E Z+, let

a; = .Sup {A(Mr) : M, is a Lipschitz constant of 172,330},
in R

139

and let
A0 = lim sup (3) .

l—ooo I

Then 2:0 f ("l (x0) / (n!) (x — x0)" converges strongly to f (x) for all x E I (a, b) sat-
isfying

A(x — $0) > A0.

Proof. Let x E I (a, b) be such that A(x — x0) > A0 and let l E Z+ be given. By the

remainder formula, Corollary 5.8, we have that

l (n) 5’30 1+1
f(x) = :0 172—10: — 2:0)" + ntmnz — $0) ,

where

A (T1($0, 13)) _>_ A(Ml)

for all Lipschitz constant M; of FL”; and hence

A (r¢(x0,x)) 2 at.

We need to show that
111210 (T‘z(1‘0, 1:)(113 — Toy-H) = 0.
Since A(x — x0) > A0, there exists t E Q+ such that
A(ID — $0) — t > A0.
Hence there exists N E Z + such that
A(x—x0)—t> T- for alllz N.

Hence

az+lA(x—xo) > lt for alllZ N.

140

Thus,
A (rl(xo,x)(x — xo)') > it for all I _>_ N,

which entails that
[1330 (rz(x0, x)(x — xo)l) = 0.
Hence
1E12(T1($0,$)($ — x0)l+1) = (x — 1E0)ll_l+123(1“1($0,$)($ — xo)l) = 0.
This finishes the proof of the theorem.

The following result is a generalization of the corresponding result about power
series with real coefficients, which was proved in [5]; and the arguments in the proof

are very similar to those in the proof of the previous result.

Theorem 5.20 Let x0 E R be given, let (an) be a sequence in R, let

A0 = lim sup { —A1(an)} ;

 

and for all n 2 0 let bn = d""°an. Suppose that the sequence (b,.) is regular; and
write U;‘,'i__osupp(bn) = {q1,q2,...} with le < qj, if jl < jg. For all n 2 0, write
bn = 2;; bnjd‘b' where bu]. = bn [q,-]; and let

 

n - 1 in R. (5.5)

— sup (limsup,H00 Ibnjll/n :j Z 1}
Then, for all a E R satisfying 0 < o < 17, the function f : ]xo — od"°,xo + ad’s] —+

R, given by
f ($) = Z an($ - $0)",

n=0

is infinitely often difi‘erentiable on ]xo — od*°,xo+od*°], and the derivatives are

given by

f(k)($)=gk($)= inm-1)---(n—k+1)an(x—x0)"-k

n=k

141

for all k 2 1 and for all x E [x0 — od"°,xo + ad’\°]. In particular, we have that

f 0‘ )($o)

k! forallk= 0,12,.

ale:
and hence for all x E [x0 - od’\°, x0 + odAO] , we have that

f(x)=Zf—f—(Lnao) (317—170)-

Proof. By a remark made at the beginning of the proof of Theorem 4.12, we may

assume that A0 = 0, bn = an for all n 2 0, and
min (Uiosupp (an)) = 0-

Using induction on k, it suffices to show that the result is true for k = 1. Since
lin1,,_.oo(n)1/"- 1 and Effioan (x — x0)" converges weakly for x E [x0 — o,xo + a],
we obtain that Z°°__1na,, (x — x0)"-1 converges weakly for x E [x0 — 0, x0 + 0]. Next
we show that f is differentiable at x with derivative f’ (x) = g1 (x) for all x E

[x0 - 0, x0 + a]; by Theorem 5.15, it suffices to show that

f(x+h)—

h f(x)—91(2)) <d-1lhl

 

for all x E [x0 - o,xo +0] and for all h 7h 0 in R satisfying x+h E [x0 —o,xo+o].
So let x E [xo—o,xo+o] be given and let h ah 0 in R be such that x + h E
[x0 - o,xo + a]. First let ]h] be finite. Since f (x), f (x + h) and 91 (x) are all at

most finite in absolute value, we obtain that

,(f(m+h;—f(x) _glm) 20

On the other hand, we have that

A (d-1 |hl) = A (ct-1) + A(h) = —1 + 0 = —1.

142

Hence

f($+h)-f($)

h — g1 (x) < d’1|h].

 

Now let |h| be infinitely small. Write h = hod' (1 + h) with ho E R, 0 < r E Q
and 0 S |h1| << 1. Let s S 2r be given. Since (an) is regular, there exist only finitely

many elements 1n [0, s] H LJ,,_Osupp(an ); write

[01 S] n Uzo=OSUPP (an) 2 {01.3, q2,8r ° ° ' rqj,8} -

Thus,

f(z+h)[s] = (fawn—x01") [3]

n=0

CD

= Z (an($+ h — 330)") [3]

11:0

n=0 (:1

= f: (213a,. [01,8] (x + h - $0)" [8 - (Ital)

= :(:%[stl (x+h-Io)" lS—CIzsl)

(:1

= i (2‘1“ M Z (ix—in (n— h" “a” ’ 220).-.) [S _ “l

l=1 11:0

22:0 an [m,n] (23 "' $0)" [3 — ql,sl
+ 2311 nan [(11, .l (h (:v- xo)"’l) [8 - ql,s]
+ 2:24—2 " 24—12" 0an sl (’L2 (33- $0)" 2) [8 - 41.8]

H
~
ll Mu.
H

Other terms are not relevant, since the corresponding powers of h are infinitely smaller
than (1‘ in absolute value, and hence infinitely smaller than db?“ for all l E {1, . . . , j} .

Thus

n=0 l=1

f (w + h) [S] = i (2]: an [m,sl (13 - mo)" [8 - (Ital)

143

J

nan [ql,s] (h (x — xo)""1) [8 - (ml)

(=1

+;(

n=2 (=1

+ f: (2]: Ping—11% [Qtsl (’12 (93 - $0)n_2) [3 — gl,s])

= f: (a. (a: — ms") is] + :3, ("hen (e - rot“) [8]

n=0

1, n: (when. (x — m”) [s] -

Therefore, we obtain that

f($+h)-f($)
h

‘91 (517) =r hofin—(fl—Z'l')‘

n=2 2

 

an (x — xo)"—2 . (5.6)

Since A (an) 2 O for all n 2 2 and since A (x — x0) 2 0, we obtain that

A(i —‘———)a«e- W) 2 0-

n=2

Thus, Equation (5.6) entails that

 

 

A(f(e’lfirhlll'fl‘fl —gl(:1:)) Zr=A(h) >A(h)—1=A(d‘1|hl);
andhence
f($+hl)1_f(x) ‘91($) << d_llhl-

 

 

This finishes the proof of the theorem.

Theorem 5.21 (Reexpansion of Power Series) Let x0 E R be given, let (an) be

a regular sequence in R, with

A0 = limsup {ingfl} = 0;

n—soo

and let 1] E R be the radius of weak convergence of f (x) = 2:10 an (x—xo)",

given by Equation (5.5). Let yo E R be such that lift (yo - 1170)] < n. Then, for

144

all x E R satisfying |§R(x — y0)| < r) - IER(y0 — 230)], we have that the power series

2:10 f(k) (yo) / (k!) (x — y0)k converges weakly to f (x), i.e.

Ttx—yo>’°=f<m)=§ian<e-eo>"-

n=0
Moreover, the radius of weak convergence of 2:10 f 0‘) (yo) / (k!) (x -— y0)k is exactly

71 - M(yo - $0)l~

Proof. Let x E R be such that Ni (x — y0)| < n — IER (yo — xo)|. By Theorem 5.20,

we have, since |§R (yo — xo)| < n, that

f(k)(y0) = inm—1)---(n—k+1)a..(yo-xo)""‘ for allkZO.

n=k

Since lift (x — yo)| < 7) - Ill? (yo - $0)|, we obtain that

“RUE-$0M = l§R($—y0+yo-$0)l=lER($—yo)+§R(3/o-$o)l
< |m($-y0)|+l§li(y0—$0)l

<17.

Hence 2:10 an (x — x0)" converges absolutely weakly in R. Now let q E Q be given.
Then

= (i: an (a: - 330)") [(1]

an (310 — $0 + $ — 90)") [(1]

fl

(
___ (i an X ( 7,: ) (yo — xoln—k (a: - 110)") [q]
(

n=0 k=0

ZZ—T—)(1(x-yo)")rq1

145

 

= (i Zn: n(n — 1) “1;!(71 — k +1)an(y0 _ x0)” (x _ 110)") [(1]

n=0 k=0

= f: i (n01 — 1) . .k!(n — k +1)an(y0 _ x0)n—k (x _ y0)k) [q]. (57)

n=0 k=0

 

Because of absolute convergence in R, we can interchange the order of the sums in

Equation (5.7) to get

 

f(x)[ql = if f: (” l" ‘ 1) ' “k,l” " k + 1)an(y0 — ear” (:1: — you) in

k=0 n=lc

= (NZ-la-(inM—1)...(n—k+1)an(yo—xo)n-k)($-y0)k) lQl

k=0 ' 12:]:

(i f(k) (yo)

k=0 k!

(I - yolk) [‘1]-
Thus, for all q E Q, we have that

(:3 fm (yo)

kl (x — y0)k) [q] converges in R to f (x) [q] .
k=0 °

Now consider the sequence (Am)m>1, where for each m,

m (k)
Am = 2%)(3—y0)k.
k=0 '

Since (an) is regular and since A (yo - x0) _>_ 0, we obtain that the sequence (f (k) (yo))
is regular. Since, in addition, A (x — yo) 2 0, we obtain that the sequence (Am) itself
is regular. Since (Am) is regular and since (Am [q]) converges in R to f (x)[q] for all
q E Q, we finally obtain that (Am) converges weakly to f (x); and we can write

00 (1:) oo

.23. ——f ,,Ey") (a: — yolk = f (x) = z a... (a: — mo)”

for all x satisfying [ER (x — yo)l < 17 — Ill? (yo — 1150)].

Next we show that 17 - [5)? (yo - xo)| is indeed the radius of weak convergence of

22:0 f(kl (yo) / (k!) (x — yo)k. So let r > 77- I9? (yo — 330)] be given in R; we show that

146

there exists x E R satisfying [ER (x — yo)| < r such that 2210 f(k) (yo) / (kl) (13 - 310)"

is weakly divergent in R. Let

 

 

 

 

 

 

 

 

 

 

310 + fizz-1319040“ if yo 2 5'30
x :
yo _ r+rI-l9?2(90"°)l if 310 < 930
Then
r + " ill _ x
|§R($_y0)]=]x-yo[ = 77 l2(y0 0)l (7'
But
ER (110 — me) + r+n_]Réyo-$0)l if yo 2 2:0
32 (x — $0) =
112(3/0 — $0) - ”flay—$0M if yo < $0
lift (90 - $0)l 'l‘ r+n-méyo—$O)l if yo 2 $0
— lift (yo - $0)l — Hen—[315110401 if yo < $0
r+n+lfiéyo-$°ll if yo 2 $0
_r+q+inéyo-eo)l if yo < $0
and hence
m _
”f(x—1130)] = T+n+| 2(3/0 330)] > 77

Hence 233:0 an (x - x0)" is weakly divergent in R. Thus, there exists to E Q such that
(2:10 an (x — x0)") [to] diverges in R. Hence (2210 f (k) (yo) / (k!) (x — yo)k) [to] di-
verges in R; and it follows that 22:0 f (k) (yo) / (k!) (x — yo)k is weakly divergent in R.

So 7) — |§R (yo — xo)| is the radius of weak convergence of 22:0 f 0“) (yo) / (k!) (x — yo)k.

In Chapter 6, we will study a large class of functions that are given locally by

power series; and we will prove more results about power series.

147

5.4 Intermediate Value Theorem and Inverse Func-
tion Theorem

Notation 5.2 Let D1, D2 C R and let f1 : D1 —1 R and f2 : D2 —> R. Then we say

that f1 ~ f2 if and only if there exists n E Z+ such that

%|f1(x)l g [f2(y)| _<_ n|f1(x)| for all x e 01 and for all y 6 D2.

Definition 5.12 Let a < b be given in R, and let f : [a, b] —> R be difi'erentiable.
Then we say that f is quasi-linear on [a, b] if and only if

SM ~ SL5 and 52,, ~ 52,5 for all x,x E [a, b], (5.8)

where 51,: and 52,3 denote the first and second secants of f at x, respectively.

Remark 5.2 It follows directly from Definition 5.12 that if f is quasi-linear on [a, b]

and if a S a1 < b1 S b, then f is quasi-linear on [a1,b1].

Lemma 5.11 Let a < b be given in R, and let f : [a, b] —+ R be quasi-linear. Then

there exist n,m E Z+ such that

gyms—nor

 

 

 

 

 

 

 

 

 

 

 

 

1|f(b) - f(a)| f(y) - f($)
H b—a Si y—x b—a ’ (5.9)
[fly] : g(x) — f(x) 5 m'flgjj)?“ Iv - ml (5.11)
for all x and for all y aé x in [a, b].
Proof. Since f is quasi-linear on [a, b], there exists n such that
1 f(x?) — f(ir) < f(y) — f(x) < n f(y) - f(x?) (5 ,,,
n y—x — y—x — y—x '

 

 

 

 

 

 

148

 

 

 

and
}_ |f(17) - f(i) - f’($)(37 - i)l < |f(y) - f( )- f’($)(y - aI)l
n (t7 - :2)? ‘ (y - w)2
nlfh?) — f(i) — f’($)(z7 — in
S (g _ :13)? (5.13)

for all y aé x and for all y 74 5: in [a, b]. Letting 5: = a and y = b in Equation (5.12),

we obtain Equation (5.9). Since

 

fl(x) = lim f(y) — f(x)

y-rz y—x

for all x E [a, b], Equation (5.10) follows readily from Equation (5.9).

Equation (5.13) entails that

f(y) - f($) |f(b) - f(a) - f'(a)(b - all
y—x Sn (b—a)2 ly—xl

for all x and for all y 75 x in [a, b]. Using Equation (5.9) and Equation (5.10), we

 

- f'($)

 

 

 

 

 

have that
lf(b)-f(ztb):af)’2(a)(b-a)l S bia(lf(hl))::(a)l+|f.(a)l)
_ |f(b)-f(a)|
— (n+1) (b—a)2
Thus,
f(y) - f(iv) |f(b) - f(a)|

 

 

y—x _fr(g;) Sn(n+1) (b—a)2

for all x and for all y 7h x in [a, b].

Iy-xl

Corollary 5.9 Let a < b be given in R and let f : [a, b] —» R be quasi-linear on [a, b].

Let n be as in Lemma 5.11. Then
1 1 1 _ 1 _
g If (w)| S If (w)| S "9 If (w)| for all 2.x 6 [a,b]; (5.14)

and hence f’ ~ f’.

149

Proof. Let x, it E [a, b] be given. Using Equation (5.10), we have that

 

 

 

 

i no] : 5““ _<_ W“ S ”no; : :(a)| and (5.15)

Combining the results of Equation (5.15) and Equation (5.16), we obtain Equation

(5.14).

Corollary 5.10 Let a < b be given in R and let f : [a, b] —> R be quasi-linear. Then

 

 

f(y)-f(x) ... f(ID—33a)forallyré$inla15l1and (5.17)

 

y—x b—
f’(x) ~ f(bb : g(a) for all x E [a, b]. (5.18)

Lemma 5.12 Let a < b be given in R, and let f : [a,b] —+ R be quasi-linear. Then

f is continuously difi'erentiable on [a, b].

Proof. Let m E Z+ be as in Lemma 5.11, and let x # y in [a, b] be given. Then

 

 

 

 

my) — f’(:v)| s “y; : if”) — f(y)] + “321“) — f’(rv)
lf(b) _ f(a)l ]y _ £13] + mlf(b) — f(all

S m (1-1)2 (1....)2 0...]

 

_ lf(b)-f(a)l
-— 2m (b—a)2 ]y—x].

Hence
f’(y) - f’(:1.-)

S 2m|f(b) — f(a)l for all y 7E x in [a, b].
y —x

(b - 0)?

 

 

 

 

Therefore, f’ is continuous on [a, b].

150

Lemma 5.13 Let a < b be given in R, and let f : [a,b] —> R be quasi-linear. Then

for all x,y E [a, b], we have that

Mf(z))—f(x))2A(flbl‘flal)“(y-x).

 

 

b—a b—a

Proof. Let x, y E [a, b] be given. Using the proof of Theorem 5.12, we have that

f (y) = f (x) + f' (50) (y - 1‘) + 7‘ ($.31) (11 - 2:)2 and (5-19)

f (x) = f (y) + f’ (y) (a: — y) + r (am) (y -— 102. (5.20)

where, using Equation (5.11), we have that

MW), 2 ,(rtb)—f<a))=,(f<b)—f(a>)_,(,_a)and

 

 

 

w—af b—a
A(rrmn 2 A(flb;:£(“))—A(b—a).

Adding Equations (5.19) and (5.20), we obtain that

(f’ (y) - f’ (22)) (y - $) = (7" ($131) + 7‘ (11.13)) (31 - $)2;
and hence
f' (y) - f' ($) = (7‘ ($.31) + My, 10)) (y - 1')-
Thus,

A(f’(y) - f(x)) = A(r(x,y)+r(y.x)) +A(y-x)

> A(f(bl)):cfl(a)) —A(b—a)+A(y—-x)

 

= A(f(bl))::(a)) +A(%:—:).

Lemma 5.14 Let a < b be given in R, and let f : [a, b] —> R be quasi-linear. If
f (a) = f (b), then f is constant on [a, b].

151

Proof. Let a: E (a, b] be given. Then, by Equation (5.17), we have that

f(x)-f(a) ~ f(b)-f(a) =0,

x—a b—a

 

 

which entails that f (as) = f (a).

Definition 5.13 Let a < b be given in R, and let f : [a, b] —+ R be quasi-linear and

nonconstant. Define g : [0,1] -* R by

_f((b-a)x+a)-f(a)
g(x)“ f(b)-f(a) '

Then 9 will be called the scaled function of f on [0,1].

Lemma 5.15 Let a < b be given in R, let f : [a, b] —> R be quasi-linear and noncon-
stant, and let g be the scaled function of f on [0,1]. Then 9 is quasi-linear on [0,1],
with

A (g (33)) = A(x) 2 0 and A (g' (22)) = 0 for all a: 6 [0,1] .

Proof. Using Theorems 5.8 and 5.9, we obtain that g is continuous on [0,1]. Using

Theorems 5.13 and 5.14, we obtain that g is differentiable on [0, 1], with derivative

, _ b—a
g (“3) ‘ f(b)-f(a)

Using Equation (5.18), we have that

 

f'((b—a)a:+a) for a1le [0,1].

 

f'((b—a)x+a)~ f(bl)’:£(a) for a1le [0,1].
Hence
g'(:1:) ~ 1 for all a: 6 [0,1].

Moreover, for all x E [0, 1], we have that

f((b-a)$+a)-f(a))
f(b)-f(a)

 

Aw» = A(

152

_ f((b-a)x+a)-f(a) (b-a):c
‘ "( <b—a>a: f(b)-f(a))

= A(f((b—c(zl)):::)a;—f(a))+A(f(bl;:cfl(a))+x\(x).

 

 

Using Equation (5.17), we have that

A(f((b-C(11))‘:*C;)“]’Mz))= A (“2:5”) = “*(nbijflmd

 

 

 

for all a: E [0, 1]. Hence

A(g(:1:))=/\(a:) ZOfor alle [0,1].

Now let x,:Z‘ 6 [0,1] be given, let SM and 51,5, denote the first secants of g at a:
and ii: and let T1,(b_a)x+a and T1,(b_a),-,+a denote the first secants of f at (b — a)x + a

and (b - a)i‘ + a. Then for all y 74 a: in [0, 1], we have that

_ g (y) - 9 (Ir)
51,1:(y) y _ SE

= 1 f((b—a)y+a>—f(<b-a>x+a)
f(b)-f(a) y—x

b—a f((b—a)y+a)-f((b—a):v+a)
f(b)-f(a) ((b—a)y+a)-((b-a)$+a)

= flab—WM“ _ aly + a)'

Similarly, for all 3‘] 76 :7: in [0, 1], we have that

 

 

 

b—a

f(b) - f(a)

Since f is quasi-linear on [a, b], we have that

51.07): T.,(._.,.+a((b_a)g+a).

TI1,(b—a):1:+a N T'1,(b—a):E-+—a-

Hence

31,: N 51,5.

153

Finally let Sgfl and 52,5, denote the second secants of g at a: and ii: and let T2,(b-a)x+a
and T2,(b_a)5+a denote the second secants of f at (b — a)a: + a and (b -— a):1': + a. Then
for all y # :1: in [0,1], we have that

 

SW) = 9<y>-9<x>—¢(x)<y—x)

 

(y - $)2
(b - a)2 .
f (b) - f (a)

f((b-a)y+a) -f((b-a)$+a) -f’((b-a)$+a)(b-a)(y-$)
((b - a)(y - 33))2

MT2.(b—a)x+a((b " a)?! + al'

Similarly, for all 37 7e 5: in [0, 1], we have that

(b - a)2

52,5(37) = W

T2,(b—a):i+a((b - a)? + (1).

Since f is quasi-linear on [a, b], we have that

T2,(b—a)1:+a ~ T2,(b—a)§:+a-
Hence
S2,: ~ 32,5.
This finishes the proof of the lemma.

The following result follows immediately from Lemma 5.15 and Lemma 5.13.

Corollary 5.11 Let a < b be given in R, let f : [a, b] —> R be quasi-linear and
nonconstant, and let 9 be the scaled function of f on [0,1]. Then for all x,y 6 [0,1],

we have that

A(g’(y)-g’(x)) Z A(x/fir)-

 

Lexnma E

stant. and

by

771611 93 i

Proof. Si

Now let )

Thus,

QM

which eDt

Since A {g

NeXt 1

154

Lemma 5.16 Let a < b be given in R, let f : [a, b] -§ R be quasi-linear and noncon-
stant, and let 9 be the scaled function of f on [0, 1]. Let 9;; : [0, 1] O R -—> R be given
by

912(X)=9(X)[0l-

Then 93 is continuously differentiable on [0, 1] H R, with derivative
(gn)’ (X) = g’ (X) [0] 7e 0 for all X e [0, 1] n R.

Proof. Since 9 is quasi-linear on [0, 1] by Lemma 5.15, there exists n E Z+ such that

]9(y) - g(x)

y_$ -9'(:v)

 

S n|y — :z:| for all y 7Q :c in [0, 1].

Now let X E [0, 1] 0 R be given. Then

 

 

 

 

 

 

]g(Y}Z:£)I((X) —g’(X) gnIY-Xl foralleéXin [0,1]flR.
Thus,
9R(Yy):§:(xl _gI(X)[0]] = ]9(Y)l;)/l:§((xllol _gI(X)[0]]
= [(902 : fix) — g' (X)) [01]

S 2nlY—Xl for all Y #X in [0,1] flR,
which entails that 93 is differentiable (in the real sense) at X with derivative
(QR), (X) = 9' (X) [0] 75 0.
since A (g’ (X)) = O by Lemma 5.15.

Next we show that (gR)’ is continuous on [0, 1] H R. As in the proof of Lemma

5.12, we have that

Ig’(y) - 9’($)I S 2nly - ml for all at. y 6 [0.1]-

155
In particular,
|g'(Y) — g'(X)| g 2n|Y — X| for all X,Y E [0, 1] H R.
It follows that
I(gn)’(Y) - (gn)’(X)l = l9’(Y)[Ol - 9’(X)[0]|
S 3nIY—Xl for all X,Y E [0, 1] 0R,

which entails that (gR)’ is (uniformly) continuous on [0, 1] n R. Thus, gR is continu-

ously differentiable on [0, 1] n R.

Lemma 5.17 Let a < b be given in R, and let f : [a, b] —> R be quasi-linear and

nonconstant. Then f is strictly monotone on [a, b].

Proof. Let g : [0, 1] —+ R be the scaled function of f on [0,1]. We show that g is
strictly increasing on [0,1]. Let gR be as in Lemma 5.16. Then g; is continuously
differentiable on [0, 1] n R and (gR)' (X) 7é 0 for all X E [0, 1] HR. Thus, gR is strictly
monotone on [0, 1] n R. Since 93 (0) = 0 < 1 = g}; (1), we obtain that 93 is strictly
increasing on [0, 1] n R. Now let x,y 6 [0, 1] be such that x < y, and let X = 9% (:12)

and Y = in (y). As a first case, assume that X < Y; then gR (X) < 93 (Y). Hence

g(y) - g(x) = on (Y) - 91200
+ (g (y) - g (1’)) + (9 (Y) - 912m)

+ (9300 - g(x)) + (g(x) - g(x)).
where the first term is positive and real. By Theorem 5.12, we have that
g(y) - 90’) = 9'(Y)(y - Y) + T0”. y)(y - Y)?

where

A(9'00) = 0, My - Y) > 0, and A(TO’, 31)) 2 0.

156

Hence |g(y) — g(Y)| is infinitely small. Similarly, |g(X) — g(x)] is infinitely small.

Since
/\(9(Y)) Z 0 and 9120/) = g(Yllol,
we obtain that |g(Y) —- gR(Y)| is infinitely small. Similarly, |gR(X) — g(X)| is infinitely
small. So
g(y) - g(x) z 9120’) - 912(X ) > 0; and hence 9 (It) < 9 (11)-
As a second case, assume that X = Y. Then y — :1: << 1, and hence

gun-g(x) = 9’(II¢)(y-~'IJ)~i~7‘(1=,y)(y--76)2

22

g' (x) (y — x), (5.21)
since Ir (2:, y)] is at most finite and hence
,\ (rm) (y — a2) = A(r(x,y)) + 2m — x)
> 2.x (y — :13)
> A(y—x)=Mg’(x))+A<y—x>
= A(g'os) (y — x».

By Corollary 5.11, we have that
A(Q'CE) -9'(X)) 2 A(x-X) > 0-

Since
g'($)~1, 9'(X)~ 1 and |g'(x) - 9'(X)|<<1.
we obtain that
9’ (x) z 9’ (X) z (919' (X ) > 0- (522)

Born Equations (5.21) and (5.22), we obtain that g (y) — g (x) > 0. Thus,

g(x) <g(y) for allx<yin [0,1];

157

and hence g is strictly increasing on [0, 1]. Since

(I:
b—

 

a
f(x) = (f(b) — f(a))g( a) +f(a) for an x 6 M
and since 9 is strictly increasing on [0,1], we obtain that f is strictly increasing on

[a, b] if f (a) < f (b), and f is strictly decreasing on [a, b] if f (a) > f (b).

The following theorem generalizes the intermediate value theorem which was dis-
cussed in [5] and which applied to functions whose domain and range are both finite
and whose derivatives are finite everywhere. We offer two proofs; and after scaling,
the second proof is similar to that of the previous version of the intermediate value

theorem.

Theorem 5.22 (Intermediate Value Theorem) Let a < b be given in R, and let

f : [a, b] —> R be quasi-linear. Then f assumes every intermediate value between f (a)

and f (b).

Proof. If f (a) = f (b), then f is constant on [a, b] by Lemma 5.14, and there is
nothing to prove. So we may assume that f (a) aé f (b). Let g : [0, 1] —» R be the

scaled function of f on [0, 1]. For all a: E [a, b], we have that

:c—a

 

f(x) = (f(b)-f(a))g(

= l209011017),

)+f(a)

b—a

where l1 and l2 are linear functions. Hence it suffices to show that 9 assumes every

intermediate value between 9 (O) = O and g (1) = 1. We present two proofs.

First proof (by iteration): Let 9;; be as in Lemma 5.16, let S 6 (0,1) be given, and
let 5;; = R (S). Then SR 6 [0, 1] n R. Since 93 is continuous on [0, 1] n R by Lemma
5.16, there exists X E [0, 1] n R such that gR (X) = SR. Thus,

IS - g(X)| S IS - Sal + lSR - 912(Xll + |912(X) - g(Xll

= IS— Sal + lgn(X) -g(X)l

 

is infinite
Now I
assume t.

g(X-l-I)

Then, by

and henc

where ]r(

Where

Let.

158

is infinitely small.

Now let 3 = S — g(X). If s = 0, then g(X) = S and we are done. So we may
assume that s aé 0; we try to find :5 E Rsuch that 0 < |:1:| << 1, X+x 6 [0,1] and
g(X+x) = 5'. Let a1 = 1/g’ (X), and let

8

g’(X)'

 

$1 = 0.18 =
Then, by Theorem 5.12, we have that

g(X+:c1) = g(X+?-(—)?)-) =g(X)+s+r(X,X+x1)xf

= S+r(X,X+:I:1)a:¥;
and hence
|g(X+x1) -S| = IT(X,X+$1)|$¥.

where Ir (X, X + 1:1)I is at most finite. So

2

 

 

 

 

s
|g(X+$1)—Sl=lT(X,X+$1)l—=0132,
(9’00)2
where
|r(X,X+a:1)|
Cl: 2
(9’00)
Let
sl=S—g(X+x1)=—T(X,X+xl)x¥,
let
a __r(X,X+a:1)a'f
2_ 9'(X‘HCI)
and let
a: -:1: +———8i—
2‘ 1 g’(X+a:1)°
Then
r(X,X+xl):z:f 1‘(X,X+:l:1)aff2 2
a: =x - = — 3 =11: +as.
2 ‘ g(XMl) 1 g(le) ‘ 2

159

Since 9’ (2) ~ 1 for all z E [0, 1] by Lemma 5.15, we obtain that |a1| and lag] are both

at most finite, and :32 z 11:1. Moreover,

X l X l | 31

= g(X+:z:1)+31+1'(X+a:1,X+ar:2)(:1:2—:z:1)2

= S+r(X +$1,X+$2) a334,
where |r (X + 2:1,X + 1132) | is at most finite. Let
Cg = ]7‘ (X +131,X +$2)|a§.
Then 62 is at most finite and
lg (X + 3:2) — S] = c234.
By induction, we obtain a sequence (xn) such that for all n 2 1, we have that
n '—l n
x" = Zajs” % als = 3:1 and |g(X +1:,,)— S] = (3,32,
i=1

with

A(an) ZOandA(c,,) ZOfor alan 1.

Since Isl << 1 and since A (Cu) 2 O for all n _>_ 1, we have that
”13.10 cusp = O; and hence “119209 (X + as") = S.
Also, the sequence (93,.) converges strongly. Let

oo

a j_1

x: hm 2:":2 ajs2 .
i=1

Then

 

160

We show that g (X + x) = S. Since 9 is continuous on [0, 1] and since the sequence
(xn) converges strongly to x, we obtain by Lemma 5.5 and Lemma 5.1 that the

sequence (9 (X + xn)) converges strongly to g (X + x). Hence

g(X+x) =JL11309(X+xn) :5.

Finally we show that X+x 6 (0, 1). First assume that X = 0; then S > 0 = g (X)

and hence s = S — g (X) > 0. Since 9’ (O) z (gR)' (0) > O, we obtain that

s
X+x=x%—>O.

9’ (0)
Moreover, x << 1; hence X + x = x 6 (0,1). Now assume that X = 1; then

S < 1 = g (1) and hence s < 0. It follows that

xz-L-<OandhenceX+x=1+x<l.

9’ (1)
Since IxI << 1, we obtain that X+x = 1+x 6 (0,1). Finally assume that 0 < X < 1;

then X is finitely away from 0 and 1. Since IxI << 1, we obtain that X + x E (0, 1).

Second proof (using the fixed point theorem): Having found X 6 [0,1] 0 R such
that g}; (X) = SR, we proceed to find x such that O < IxI << 1, X + x E [0, 1] and

g (X + x) = S. Since 9 is differentiable on [0,1], we have, by Theorem 5.12, that
S=g(X+x)=g(X)+g'(X)x+r(X,X+x)x2, (5.23)

where Ir (X, X + x)| is at most finite.

Transforming Equation (5.23) into a fixed point problem yields

3 _r(X,X+x)$2
9’(X) 9’(X)
M93).

 

(5.24)

where s = S — g (X) is infinitely small in absolute value. Let

M={z€R:/\(z)2)\(s)}.

161

Let x E M be given. Since Ir (X ,X + x)| is at most finite and since g’ (X) ~ 1, we

have that

,\ (($653?) _>_ 2A(x) > A(x) 2 A(s) = A (3,575).

Hence 3 / g’ (X) is the leading term on the right hand side of Equation (5.24). Thus,

 

8

h(x) z m; and hence A (h (x)) = A (s) for all x E M.

Hence

h(M) c M.

Now let x1 76 x2 be given in M. Then

r(X,X+x2)x§-r(X,X+x1)xf
9’(X)

= |g(X+x2)-9(X+x1)
9’(X)

 

Ih($1) — h(x2ll =

 

 

 

+171—1132 .

 

But
g(X—l-xg) =g(X+x1)+g'(X+x1)(x2 —x1)+r(X+x1,X+x2) (xg—x1)2,
where Ir (X + x1,X + x2)I is at most finite. Thus,

lh(xi) - h(51:2)l

 

 

 

 

 

 

 

___ |g’(X+$1)($2-$1)+7'(X+$1,X+$2)($2—$1)2+3: —x
#00 1 2
_ gI(X+$1)—g,(X) x —x r(X+x1,X+x2) x —x 2
‘ I am (2 1” 9'00 (2 1)
_ |9’(X+$1)-9'(X)| |T(X+$1,X+$2)I _
S I181 $2] , le 932] -
9’00 9 (X)

Using Corollary 5.11 and the fact that g’ (X) ~ 1, we have that

lgl(X+$l)-9I(X)I _ I (I: _ I
A( 9,0,) ) — A(g(X+1) g(X))

 

Z A(x1)_2 A(s)
A(s)
2 .

 

162

 

 

Also
Ir(X+x1,X+$2)l —x x —x
"( 9’06) '2" 2') Z A“ 2)
_>_ min{A(x1),A(x2)}ZA(s)
> Aés).
Hence

Ih(.’L‘1) — h(x2)I << dA(s)/2 I121 — 232] ,

where A (s) > 0. So h is contracting on M, and hence h has a fixed point x in M.

The same argument as at the end of the first proof shows that X + x E (0, 1).

The following two examples show that the conditions in Equation (5.8) are nec-

essary to obtain Theorem 5.22.

Example 5.10 Let f : [—1, 1] -+ R be given by

f(x)={d if0_<_|xI<<1

x3 ifx~1

Then f is continuous on [-1,1] since

lf(y)-f(w)
y—x

 

S 3 for all x 79 y in [—1,1].

 

Next we show that f is differentiable on [—1, 1] with derivative

f’(x)=9(x)={ 3x2 if0< |x|<<1

 

 

if x ~ 1
Let x 75 y be given in [—1, 1]. We show that
x
If(*’]_ f() —g<x> <3ly—wl-
First assume that 0 S IxI << 1. Then

 

 

I f f($)_

 

I f f___(x)

 

o ifOS|y|<<1
Isl-AI ify~1 '

y—x

163

But for y ~ 1, we have that

y3-d
y—IE

y3-d
y—x

 

 

zy2SIyIzIy—xl;andhence <2Iy—xI.

 

 

 

 

Thus, for O _<_ IxI << 1 and for all y 7é x in [—1,1], we obtain that

If(y;:£(x) -g(a:) <2Iy—xl.

 

Now assume that x ~ 1 and 0 _<_ IyI << 1. Then

If(y)-f($)_ (
y—x

 

 

13)

 

 

22
to
H
N
/\
N)
E
l
[\D
E—
I
313.

and hence

 

f (y) - f (x) _
y - x
Finally, assume that x ~ 1 and y ~ 1. Then

f(y)-f(x)
y—(L'

 

3 3
_ y '3
g(x)] 3H,

= I(y+2x)(y-:v)|=ly+2xlly-x|

— 3x2

 

= Iy2 + xy — 2x2]

 

 

 

< 3Iy—xI.

Using Theorem 5.15, we have that f is differentiable on [—1, 1], with derivative

r 0 'f0_<_ <<1
f(x)=g(x)={3$2;fx~|1$| .

But f is not quasi-linear on [—1, 1] since f is neither constant nor strictly monotone

on [—1, 1]. f does not satisfy the intermediate value theorem on [—1, 1] since

f(—1)=—1<0<1=f(1);butf(x);é0forallx€[—1,1].

Example 5.11 Let f : [0, 1] —+ R be given by

2x ifx~1 andx[0]¢Q.

7rx ifx~1andx[0]€Q
“H
x ifOSx<<1

164

Then f is differentiable on [0, 1], with derivative

1r ifx~1andx[0]€Q
f’(x)={2 ifx~landx[0]¢Q.
1 ifOSx<<1

Thus

f(1)-f(0)

f’(x)~1= 1-0

 

for all x 6 [0, 1] .

But f is not quasi-linear on [0, 1] since f is neither constant nor strictly monotone on

[0,1]. We have that
f(O)=O<1<7r=f(1);butf(x)7é1forallx€[0,1].

In this example, the first condition in Equation (5.8) is satisfied, but the second
condition is not. We also note that even though f’ (x) is positive and finite for all
x 6 [0,1], f is not strictly increasing on [0,1], which is another manifestation of the

differences between R and R.

Using Lemma 5.17 and Theorem 5.22, we readily obtain the following result.

Corollary 5.12 Let a < b be given in R, and let f : [a, b] —> R be quasi-linear. Let
a = min {f (a) ,f (b)} and fl = max {f (a) .f (b)}- Then

f ([a,b]) = [OW]-

Theorem 5.23 (Inverse Function Theorem) Let a < b be given in R, and let
f : [a, b] —+ R be quasi-linear and nonconstant. Let a = min {f (a), f (b)} and
fl = max {f (a) , f (b)}. Then f has an inverse function f '1 : [a,fi] —+ [a, b] that is
quasi—linear on [a, 6], with derivative

1

W for allpE [a,fi].

(f‘1)'(p) =

165

Proof. That f ‘1 exists follows immediately from the fact that f is strictly monotone
on [a, b], by Lemma 5.17. Let m,n E Z+ be as in Lemma 5.11. First we show that
f‘1 is continuous on [0,6]. Let p 74 v in [0,6] be given and let x = f‘1(p) and

y = f ‘1(v). Then, using Equation (5.9), we obtain that

 

 

 

 

f"(v)-f"(p)I = I 31-2
v—p f(y)-f(x)
b-a

 

S n|f(b)-f(a)l'
Thus

f“(v)-f“(p) <71 b-a
v-p ‘ |f(b)-f(a)|

and hence f ‘1 is continuous on [a, 6].

 

 

for all p 75 v in [a,fl];

 

 

Next we show that f"1 is differentiable on [a, )8]. Let p 9L4 v in [a, ,8] be given, and
let x = f‘1(p) and y = f‘1 (v). Then, using Equations (5.9), (5.10) and (5.11), we

obtain that

 

If‘1(v) - f’1 (p) _ 1 I
v - p f’ (f‘1 (0))

_I y-x __1_
— f(y)-f(x) f(x)

 

 

1
If’ (It‘ll

If ’ (w)| (b - a)2

f(y)-f(fl?)

-$

y—x
f(y)-f(x)

 

- f’ (it)

 

 

 

y—III
f(y)-f(x)

_ mlf(b)—f(a)| y-x 2 1 _ x
‘ (1)—av (f(y)-f(z)) lf’(x)llf(y) fl“

 

 

|/\

 

 

 

_ mlf(b)-f(a)l y—x 2 1 .—
‘ (b—av (f(y)-f(x)) |f’(:c)|| p‘

< m|f(b)-f(a)ln2 <b—a)? n b
— (b-aV (f(b)-f(a))2 lf(b)

f(a)| '“ ’ ”'

 

 

166

mn3 b — a
(f (b) - f (0))

Using Theorem 5.15, we obtain that f ‘1 is differentiable on [(1, 3] with derivative

 

2lv-Pl-

"I, -———l or a
(f )(p)—f,(f_,(p))f anpe[ .Bl-

Now let p, p 6 [01,6] be given, let S”, and 51.13 denote the first secants of f‘1 at p
and ,6, let x = f‘1(p) and x = f‘1(p), and let T1,: and T1,; denote the first secants of
f at x and i. For all v 75 p and for all 17 75 ,5, let y = f’1(v) and y = f‘1('D). Then

we have that

 

 

 

 

_ f“(v)-f‘1(p)= y-z = 1 n
S”“"' v—p .nm—fo) Tnooad (5%)
_ _ f‘1(v)—f-1(fi)_ 37—5: _ 1

Since f is quasi-linear on [a, b], we have that T1,:c ~ Tm. Hence Equations (5.25) and
(5.26) entail that

51,. ~ 51,. for all p. p e [a, a].

Finally, using Corollary 5.9 and the fact that T1,; ~ T1,; and T2,; ~ T2,5 for all

x, x E [a, b], we can easily verify that
Sap ~ 52.5 for all P, fi 6 [a, 5],

where 32,p denotes the second secant of f ‘1 at p and T2,: the second secant of f at

x. Hence f‘1 is quasi-linear on [a, 6].

Chapter 6

Expandable Functions

In this chapter, we will present a detailed study of a large class of fimctions on R,
which are given locally by power series with R coefficients [41, 43]. We show that
these functions have all the desired smoothness properties; in particular they satisfy

all the common theorems of real calculus.

6.1 Definition and Algebraic Properties

Definition 6.1 Let a,b E R be such that 0 < b — a ~ 1, let f : [a, b] ——» R and let
x0 E [a, b]. Then we say that f is expandable at x0 if and only if there exists 6 > 0,
finite in R, and there exists a regular sequence (an (x0)) in R such that, under weak

convergence, f (x) = 22°20 an (330) (x — x0)" for all x E (x0 — 6, x0 + 6) n [a, b].

Definition 6.2 Let a, b E R be such that 0 < b - 0. ~ 1 and let f : [a, b] ——> R. Then

we say that f is expandable on [a, b] if and only if f is expandable at each x E [a, b].

Definition 6.3 Let a < b in R be such that t = A(b — a) aé 0 and let f : [a, b] —* R.
Then we say that f is expandable on [a, b] if and only if the function g : Id“a, d“b] —->
R, given by

g(x) = f (61%).
167

168

is expandable on Id“a,d“b].

Lemma 6.1 Let a,b E R be such that 0 < b—a ~ 1, let f,g : [a,b] —» R be

expandable on [a, b] and let a E R be given. Then f + ag and f . g are expandable on
[a, bl-

Proof. Let x E [a,b] be given. Then there exist finite 61 > 0 and 62 > 0, and there

exist regular sequences (an) and (bn) in R such that

OSIh|<61 => f(x+h)=2a,,h" and

n=0
0<|h|<62 => g( (x)+h= 2b,, h".
Let 6 = min {61/2,62/ 2}. Then 0 < 6 ~ 1. Moreover, for 0 S IhI < 6, we have that

(f+ag)(x+h) = f(x+h)+ag(x+h)

—_- Zanhn+a2bnhn
n=0 "=0

= Zanh" + Z (abn) h
n=0 n=0

= 2 (an + ab") h
n=0

where Err-0 (a7, + ab") h" converges weakly and where the sequence (a7, + ab") is
regular by Lemma 4.1. Thus (f + 09) is expandable at x. This is true for all x E [a, b] ;

hence (f + ag) is expandable on [a, b] .
Now for each n, let
on = Zn: ajbn_j.
Then the sequence (on) is regular by Lemma 4.1. Since 2°10 anh" converges weakly

for all h satisfying x + h E Ia, b] and 0 < IhI < 61, so does Zn_0a,, [t] h" for all

169

t E Unsupp(a.,,). Hence 3,10 I(a,, [t] h") [qII converges in R for all q E Q, for all
t E Unsupp(a,,) and for all h satisfying

3 3

x+hE[a,b], OSIhI<§6SZ¢51 and lhlaégb.

Now let h E R be such that x + h E [a, b] and O S IhI < 6. Then

00 oo
2: I(a,,h") IQ“ = Z 2 an [(11] h" [92}
"=0 "=0 <11 6 wpp(an). 92 6 '“PP("")
qr + 92 = 9
00
S E 2 Ian [(11]! W [call-
91 6 Ufzosupphn). <12 6 Ufi°=ouupp(h") "=0

91+92=q

Since 22:0 Iafl [q1]I Ih" [q2]I converges in R and since only finitely many terms con-

tribute to the sum 2 by regularity, we ob-

(ll 6 Uitiosurmmn). (12 6 Utiosuppw")
(11 + (12 = q

tain that 2 fin°°_0 I(a,,h") [q]I converges for each q E Q. Since 2:10 anh" converges abso-

cc
1120

lutely weakly, since 2 bnh" converges weakly and since the sequences (23:0 amh’")

and (22:0 bmh'") are both regular, we obtain by Theorem 4.9 that

Z anh" - Z bah” = Z cnh"; hence

n=0 n=0 n=0

(f-g)(:v+h) = firth".

n=0

Thus (f - g) is expandable at x. This is true for all x E [a, b] ; hence (f - g) is expand-
able on [a, b].

Corollary 6.1 Let a < b in R be given, let f, g : [a, b] -> R be expandable on Ia, b]

and let 0: E R be given. Then f + (19 and f - g are expandable on [a, b].

Proof. Let t = A(b - a), and let F, G : Id"‘a, d“b] be given by

F(x) 2 f(d‘x) and C(x) = g(d‘x).

170

Then, by definition, F and G are both expandable on [d“a,d“b]; and hence so is
F + 0G by Lermna 6.1. For all x E [d“a, d“b], we have that

(F + aG’)(x) = F(x) + aG'(x)
= f(d’IB) + 0961155)
= (f + ag)(d‘$)-
Since (F + aG) is expandable on [d“a, d“b], so is (f + ag) on [a,b].
Also by Lemma 6.1, we have that (F - G) is expandable on [d"a, d“b], where for
all x E [d"a, d“‘b],

(F - G)(Iv)

ll
:2
a
52
:1

Hence (f - g) is expandable on [a, b].

Lemma 6.2 Let a < b and c < e in R be such that b — a and e — c are both finite.
Let f : [a, b] —* R be expandable on [a, b], let g : [c, e] —+ R be expandable on Io, e],

and let f ([a, b]) C [c,e]. Then 9 o f is expandable on Io, b].

Proof. Let x E Ia, b] be given. There exist finite 61 > 0 and 62 > 0, and there exist

regular sequences (an) and (bn) in R such that

|h|<61andx+hE[a,b] => f(x+h)=f(x)+Za,,h”;and
n=1

|y|<62andf(x)+y€[c,el => g(f<x)+y)=g<f(x>)+§:bnyn.

Since F (h) = (23;, anh") [0] is continuous on R, we can choose 6 E (O, 61 / 2] such

that

Z anh" < -62—2.

n=1

IhI <6andx+hE [a,b]=>

 

 

171

Thus, for IhI < 6 and x + h E [a, b] , we have that

(9°f)(x+h) = g(f(x+h))

= 9 (f (I) + gun“)
= g(f(=r))+;b" (:mnlk

= (W +251: (Zanhny. (6.1)

=1

For each k, let Vk (h) = bk( 3;, a,,h")'c . Then Vk (h) is a power series in h

h) = Zakjhj,
j=1
where the sequence (akj) 1 is regular m R for each It. By our choice of 6, we have

jg
that for all q E Q, °°__1I(a akJ-h’ ) [qII converges in R; so we can rearrange the terms
)

in VI: 0‘) I‘ll = =1 (alcjh

verges; so (see for example [31] pp 205-208) we obtain that

[q]. Moreover, the double sum 2,311 2:? _1(ak,-hj) [q] con-

((90f f)(x+h))[q] = «gone ))[,]+)°‘:§;(ak,hj)[q]

k: =1

H
Q.

= ((90f)($ ))lq]+§§:(akjhj)[ql

j=1k=1
for all g E Q. Therefore,
(gof)(x+h) = (W x)+ZZakJ-hj
k=1j=1

= (90f)( M+ZZakJ-hj.

j=1 k=1

Thus rearranging and regrouping the terms in Equation (6.1), we obtain that

(gof)(x+h)= ()(90f113)+:czhl,

where the sequence (at) is regular.

172

Corollary 6.2 Let a < b and c < e in R be given. Let f : [a, b] —+ R be expandable
on [a, b], let 9 : [c, e] —+ R be expandable on [c, e], and let f ([a, b]) C [c, e]. Then go f

is expandable on Ia, b].

Lemma 6.3 Let a,b E R be such that O < b— a ~1, and let f : [a,b] ——» R be

expandable on [a, b]. Then f is bounded on [a, b].

Proof. Let (an (a)) and (an (b)) be the expansion coefficients of f to the right of
a and to the left of b, respectively. Let a}; = §R(a) and b3 = 92(b), and define
73 [a,b] U [012,le -’ R by

f(x)=

floan(a)(x—a)" ifx E [aR,a) .

{ f (x) if x E [a, b]
3.10 an (b) (x - b)" if x E (b.1231

Then 7 is expandable on Io, b] U [aR,bR]. For all X E [ambg] (‘1 R there exists
6 > O in R and there exists a regular sequence (an (X )) in R such that f (x) =

;‘,‘:__Oa,,(X) (x—X)" for all x E (X—6(X),X+6(X)) 0 [a,b]. We have that
{(X — 6 (X) /2,X + 6 (X) /2) n R: X E [ambit] n R} is areal open cover ofthe com-
pact real set [ambg] D R. There exists m E Z+ and there exist X1,...,Xm E

Ian, 63] O R SUCh that

[ambnl HE C 11;":1((X,-— Him},- +£¥I 0R).

 

It follows that
[a, bl U [012,le C U311 (X1 — 5 (Xi) ,X, + 5 (lelo

Let

1= min {nun{u;:°=osupp<a.(x.~»}}.

lstm

173

Then
If (x)I < d”1 for all x E [a, b] U [aR,bR] ,

and hence f is bounded on [a, b] U IaR, b3]. In particular, f is bounded on [a, b].

Remark 6.1 In the proof of Lemma 6. 3, l is independent of the choice of the cover.
It depends only on a, b, and f; we will call it the index of f on [a, b] and we will
denote it by i(f). Moreover, for all A > 0 in R and for all X E (ambg) F] R there
exists Y1 E [a,b] an(X - A,X) and there exists Y2 E Ia,b]flRfl(X,X+A) such
that AUG/1)) =i(f) = AUG/2))-

Proof. Let X1,...,Xm and I be as in the proof of Lemma 6.3. Let Z1,...,Z,c E
[ambg] n R, let {(ZJ- — 6(Zj) ,ZJ- + 6(Zj)) n R : j E {1, . . .,k}} be an open cover of
[ambn] , with 6(Zj) > 0 and real for all j E {1, . . . ,k}; and let

11 = 11312,: {In111 {Un=osuPP (an (Zj))}} -

Suppose l1 aé 1. Without loss of generality, we may assume that l < l1. In particular,

l< oo. Define f3: [aR,bR] flR —» Rby

fa(Y) =_f-(Y) [ll-

Then for Y E (X, — 6 (X3) ,X, + 6 (X,)) H [ambn] O R, we have that
fa (Y) = (in. (X.) (Y — X0") [11

= fa. 1X.) [11 (Y — X.)". (6.2)

Thus f}; is analytic on [am b3] 0 R. Moreover,

fa (Y) =70”) U] = 0

174

for all Y E (Z1 — 919,21 + Egg) (‘1 [ambg] n R. Using the Identity Theorem for
analytic real functions, we obtain that f}; (Y) = O for all Y E [aR,bR] 0 R. Using

Equation (6.2), we obtain that
an(X,-)[l] =0 for alan Oand for allj E {1,...,m},

which contradicts the definition of l. Thus l1 = l.

Corollary 6.3 Let a, b and f be as in Lemma 6.3 and let i( f) be the index of f on
[a, b]. Then

2'(f) = min{supp(f (27)) =33 6 [0,5]}-

Corollary 6.4 Let a < b in R be given, and let f : [a, b] —1 R be expandable on [a, b].
Then f is bounded on [a, b].

Definition 6.4 Let a, b, f and f be as in Lemma 6.3 and let i( f) be the index of f
on [a, b]. The function f}; : [ambg] O R —» R, defined by fR (X) = f(X) [i (f)], will

be called the underlying real function of f on [ambn] 0 R.

Remark 6.2 Let a, b E R be such that 0 < b—a ~ 1, let f : [a, b] —+ R be expandable
on [a, b], and let f}; be the underlying real function of f on Ian, b3] 0 R. Then, by the

proof of Remark 6.1, fl; is analytic on [am b3] 0 R; in particular, fR is continuous on

[am b3] 0 R.

6.2 Calculus on the Expandable Functions

In this section we show that like in the case of continuous real functions over closed
and bounded real intervals, the expandable functions over closed intervals satisfy
an intermediate value theorem, a maximum theorem, and a mean value theorem.

Consequently, the expandable functions are integrable.

 

175

6.2.1 Intermediate Value Theorem

In this section, we state and prove the intermediate value theorem for the expandable
functions, which is a generalization of the corresponding result for normal functions
[5]. We first find a real intermediate point where the real part of intermediate value is
assumed by the real part of the function; then we look for a solution in the infinitely
small neighborhood of this real point where the function is given by an infinite power
series with R coefficients. Thus, finding the point where the intermediate value is
assumed requires finding a root of a power series. We find this root by first finding
a root x0 of the leading polynomial in the power series and then applying the fixed

point theorem in the second iteration, as the proof below will illustrate in details.

Theorem 6.1 (Intermediate Value Theorem) Let a, b E R be such that 0 <
b — a ~ 1, and let f : [a,b] ——> R be expandable on Ia, b]. Then f assumes every

intermediate value between f (a) and f (b).

Proof. Let f be as in the proof of Lemma 6.3 and let f}; be the underlying real
function of f on [a R, b R] flR. Without loss of generality, we may assume that i (f) = 0.
Now let S be between f (a) and f (b). Without loss of generality, we may assume
that f (a) < O = S < f (b). Since f}; is continuous on [ambg] n R, there exists
X E [ambg] 0 R such that f}; (X) =2 0. Let ZfR = {X E [ambg] n R : fR (X) = 0},
let

0 if {ambg} n Zfa = 0

A = {a} if {012,512} I") Zia = {an}
{b} if {aRrbR} n an = {bR} ,
{a,b} if {ambg} n 2.61! = {aR,bR}

and let B = (ZfR \ {£13, bR}) U A. If there exists X E B such that f (X) = 0, then

we are done. So we may assume that f (X) yé 0 for all X E B.

176

First Claim: There exists X0 E B such that for all finite A > 0 there exists

x E (X0 — A,Xo + A) n [a, b] with A(x — X0) = 0 such that f(x) /f (X0) < 0.

Proof of the first claim: Suppose not. Then for all X E B there exists A (X) > 0,

finite in R, such that

_f_I£)_ ora x — a wi x- =
“202‘” 11 “X A()(),X+A(X))fi[,b] thM X) o. (6.3)

Since fR is continuous on [am b3] (1 R, we have that for all Y E ([aR, b3] 0 R) \ Z fR
there exists a real A(Y) > 0 such that f}; (X) /fR (Y) > 0 for all X E [ambRI fl
Rn (Y - 2A (Y

v

,Y + 2A (Y)). It follows that, for all Y E (Ian, b3] U R) \ me

Q...

33)
(Y)

 

>0for alle (Y—A(Y),Y+A(Y))fl[a,b].

Q...

In particular,

1L3)- or x — a wi x— =
f(y)>0f all 60/ A(Y),Y+A(Y))n[,b] thM Y) 0. (6.4)

Combining Equation (6.3) and Equation (6.4), we obtain that for all X E [aR, b3] 0 R

there exists a finite 6 (X) > 0 such that

Afl>0 foralle(X—6(X),X+6(X))fl[a,b]

f(x) - with A(x—X) =0, ifX E (ambg)

L122 foralleIa,a+6(X))fl[a,b]

N“) Z 0 with A(x—a) =0, ifX = an (6'5)
£13220 foralle(b—6(X),b]fl[a,b]

H
A

0"
V

with A(b—x)=0, ifX=bR

{(X — §R(6 (X)) /2,X + §R(6 (X)) /2) H R: X E [(13, b3] 0 R} is a real open cover of
the compact real set [ambn] n R. Hence there exists m E Z + and there exist

X1, . . . , Xm E Ian, bR] Fl R such that

[ambfl] r1 R c 113;, ((X, - ———m(6éxj)),xj + ___éR(6éX,-))) n R).

Thus
[a, b] C U?‘___1(Xj — 6 (X3) ,XJ- + 6 (Xj)). (6.6)

177
By Equation (6.5), we have forj E {1, . . . ,m} that

m > o for all x e (X, — 600) ,X, +6(X,-)) 0 [a,b]
f(xj) "" With A(x - X3) = 0, if Xj E (GR,bR)

[El foralle [a,a+6(X,-))fl[a,b]
1(0) 20 with A(x—Xj)=0, if ijaR (6'7)
M>0 foralle(b—6(Xj),b]fl[a,b]
f(b) _ WithA($—Xj)=0,lej=bR

Using Equation (6.6) and Equation (6.7), we obtain that f (b) / f (a) Z 0, a con-
tradiction to the fact that f (a) < 0 < f (b). This finishes the proof of the first

claim.

Since f is expandable at X0, there exists a real 6 (X0) > 0 and there exists a

regular sequence (a,, (X0))n21 in R such that

as I’ll <6<Xo> =>f(Xo+h) =f(xo)+§a.(xo>h".

n=1

Now we look for x such that 0 < IxI << 1 and f (X0 + x) = S = 0. That is we look

for a root of the equation

f (X0) + i0," (X0) :13" = 0.

n=1

Since fR (X0) = 0, we have that 0 < If (X0)I << 1. Let
m = min{n 21:A(a,,(Xo))= 0}.
Such an m exists by virtue of Remark 6.1. Consider the polynomial

P (x) = f (X0) + a1 (X0) x + - ' - + am_1(Xo)x’"—1 + am (X0) x'".

Second Claim: P (x) has a root x0 E R.

Proof of the second claim: Suppose not. Then by the thdamental Theorem of

Algebra, we have that m is even and 3&1)??? > 0. Thus

P (x)
f (X0)

 

> 0 for all x E (—6 (X0) ,6 (X0)) with A (x) = 0. (6.8)

178

There exist M1 > 0 and M2 > 0 in R such that

6 (X0) 5 (X0)
2 , 2

 

IP(x)| > M1 for all x E I— I with A(x) = 0

and

 

 

2 an (Xo) 2:"

n>m
__ . M1 ’17 5(X0)
61—mn{(2M2) ’ 2 I

Then 61 > 0, 61 is finite, and

 

< M2I30|m+1 for all x E I-6 (:(0), 6(:(0)I .

Let

 

 

 

2 an (X0) xn < M2 |$Im+1
n>m
All.
2
Af—éxl for all x E (—-61,61) with A(x) = 0.

Thus f (X0 + x) = P (x) + Ewman (X0)x" has the same sign as P (x) for all x E
(—61,61) with A(x) = 0. Using Equation (6.8) and the fact that 61 < 6(Xo), we

obtain that W > 0 for all x E (—61,61) with A (x) = 0, or

f (1’3)
f (X0)

 

> 0 for all x E (X0 —61,Xo+61) with A(x—X0) =0,
which contradicts the result of the first claim. This finishes the proof of the second
claim.

So P (x) has a root x0 E R. Since A (a,c (Xo)) > 0 for all k < m, we obtain that

A(xo) > 0. Let j E {1,...,m} be such that

A(an(Xo)x3) > A(a,(X0)xg) for alln< jand

A(an(Xo)x3) 2 A(aj (Xena) for alanj. (6.9)

179

Such a j exists by our choice of m and the fact that A (x()) > 0. We look for x w x0
such that 0 = g (x) = f (X0 + x) = P (x) + Zn)", an (X0) x". Write x = x0 + y with

A (y) > A (x0). Then, using the results of Theorems 5.20 and 5.21, we have that

9(330 + y) = 9(1130) + i 51: (Xop’lio) 31" = 0, (6.10)
k=1

where for k =1,2,...

 

°° (—n kk+1 _
(X0,IE0)= 2;” )an (X0) (133 k.

Using Equation (6.9), we obtain that A (61 (X0, x0)) = A (a,- (X0) x34) and
,\ (a. (Xo,xo)) 2 A (a, (X0) $34) for all k 2 2.

We write Equation (6.10) as a fixed point problem

y= —9($o) _ °° 5k(Xo,$o)yk
[31(Xofl70) k=2161 (meo)

where A (9 (x0)) = A (Zn>m an (X0) x3) 2 (m + 1) A (x0). Note that, by our choice of

= h (y) , (6.11)

L
A (31 (Xo,$o)) = A (a, (X0) xii—1) S A (am (X63314) = (m — 1) A (150)-
Thus A(9($0)/51(X0a$0)) = A (9 (150)) — A(51(X0,170)) 2 2A (530)-

Let

_ z , z 9(330)
M_{ ER'M)Z/\(fl1(Xo,fl¢o))}

and let y E M be given. For all k 2 2, we have that

31: (Xmil'o) k4) (,3_k___( Xo,$o)$ 3.11))
A(51(Xo,$o)y > A 51(X0,$o)$

A(flk (Xo,il?o)1130—1 —A (51 (X0, 5130))

A (a,- (X0) x(kag‘l) — A (a,- (X0) xfi—l) = 0

IV

Thus

(‘é-f—l)“ (WU—d

180

for all k 2 2. So —g (x0) /51 (Xo,x0) is the leading term on the right hand side of
Equation (6.11), and hence h(y) z —g(xo) /fll (Xo,xo). Thus h(M) C M. Now let

y1,y2 E M be given. Then
m (X —
h(M)-M312):-y1)Z(gf-—(-—X:’::;Cz:ylyf‘yy2 -))
Since A (yl) _>_ 2A (x0) and since A (y2)> 2,(Axo) we have for all k > 2 that
file (Xo,$o) ,3]: (Xoflio) 2(k-1))
A (181((X0130) (25M ”2 1-12)) A (51(Xo,$o)xo
= A (file (X01 1.0)) — A(fl1(X01xO)) + 2(k - 1) A(170)

Z A (02' (X0) Ill—k) — A (a,- (Xo) 1171—1) + (2k — 2) A ($0)

= (k " 1) A (370)
Z A ($0)
Hence
|h(y1) - h(y2)| << dflgfl lyl — ml for all 111,112 6 M.
where
at“? << 1.

So h is contracting on M; and hence, using Theorem 3.3, h has a (unique) fixed
point yo in M. Thus

f(X0+$0+y0)=0.

Remark 6.3 In the proof of Theorem 6.1, g(x) = f (X0) + Zf=1an(Xo)x" has at

mostmrootsinR,={xER:0<IxI<<1}.

Proof. Let x1, . . . ,xmo be the roots of

P (SE) = f (X0) + (11 (X0) 1' + ' ° ° + am—l (.X0)1L'm_-1 + am (X0) 23m

181

in R,. Then m0 S m. By the proof of Theorem 6.1, we have that for each l E
{1, . . . ,mo} there exists a unique m z x) in R, such that g (m) = 0. We show that
771,. . . film are the only roots of g (x) in R8. So let 77 E R, be such that g (77) = 0.
Thus

f(X0)+a1 (X0)77+"'+am(Xo)77m+ 2%(Xo)77"=0,

n>m

and hence

P(l)) = f (X0) + a1(X0)n + ' ° ° + am (X0) 77'” = — 2: an (Xo)77n- (6-12)

n>m

We look for y E R, such that A (y) > A (n) and P (n + y) = 0. Thus,

0 = P(n+y)=f(Xo)+al(Xo)(77+y)+"'+am(Xo)(77+3/)m

= P (n) + Zak(Xo.n)1/’°. (6.13)
k=1
where for k =1,2,...,m

ak(Xo,77)'-= in...(n’;!-k+1)

11:]:

an (X0) nn—k.

 

Let j E {1, . . . ,m} be such that

A(an(Xo)n") > A(aj (X0171) for alln < j, and

A(an(Xo)n") 2 A(aj(Xo)nj) for allnE{j,...,m}. (6.14)
Thus

A(01(X0,77)) = A(aj(Xo)le_l) and

A (oz,c (X0,77)) Z A (a,- (X0) nj‘k) for all k E {2, . . . ,m}. (6.15)

We write Equation (6.13) as a fixed point problem

—P(77) _ m ak(X0177)yk
a1 (X0, 7)) k=2 011 (X0, 77)
= h(y).

(6.16)

182

where

A (—P(n)) = A (2 an (X0)n") using Equation (6.12)

n>m

2 (m + 1) A (n) .
Note that, by our choice of j,

A (m (X66) = 1 (a. (X0) 76-1) s A (amour-1) = (m — 1) A (n)-

 

Thus
-P(n) _ _ _ a
A(01(Xo,n)) _ M P(n)) A( 1(Xo,n))
Z (m+1)A(77)-(m—1)A(n)
= 2A(n)-
Let

M=IZER:A(Z)ZA(ZI(J—g,)—n))}°

Let y E M be given. For all k E {2 ..... m} , we have that

A(flykl = A (wyk'l) +,\(.,)

But
ak (X0, ) _1 k—l
A (mink ) = A (at: (X0177) 77 ) " A (01 (X0170)
Z A (02' (X0) Til—”64) — A (a,- (Xo) TIL—1)
= 0.
Thus

 

 

01!: (X017?) 1: P(n)
A(a1(Xo,n)y)>A(a1(Xo,77)) for allkE{2 ..... m}.

183

So —P (77) /al (X0, 77) is the leading term on the right hand side of Equation (6.16),
and hence h (y) m —P (77) /a1 (X0,r7) . Thus

h(M) C M.
Now let y1,y2 E M be given. Then

h(yl) - h(yzl = EW (31bc - 91)

= 72-72612:(all)

Since A (yl) 2 2A (77) and since A (yg) 2 2A (77), we have for all k E {2, . . . ,m} that

O]; X(0,7]k_1__>()) (all? (X07 Tl) 2(lc—1))
A A _—
(01( (Xofl?) )(Ic231 yzy — (11 (X0177),7

= A(ak(Xo,77)) A(011(Xo, 77))+ 2(IV-1)A(77)

2 A(a,(x 177%) 1(a,(xo)ni-1)+(2k—2)1(n)
= (k-1)A(n) 2 M77)
M77)
> 2 > 0.
Hence
1121
lh(y1) " h(1/2)I << d 2 I311 - 312] for all 9113/2 E M,
where
0111211 << 1.

So h is contracting on M, and hence h has a (unique) fixed point yo in M. Thus
P(77+3/0) = 0 with M770) 2 2A(/7) > A07)-

Thus,

n+mzn

184

Since P(77 + yo) = 0, there exists I E {1, . . . ,mo} such

77+yo=$7-

Hence

7] z 1171.
By uniqueness of 771 as a solution to

{9($)=0

3

22 z x;
we obtain that
Tl = 771,

which finishes the proof of the remark.

Corollary 6.5 Let a,b E R be such that 0 < b — a ~ 1, let oz < 6 be given in
[a, b] and let f : [a, b] —» R be expandable on [a, b]. Then f assumes an [a,fi] every

intermediate value between f (a) and f (,6).

Proof. Let t = A(fl — a), and let 9 : [d“a,d“fi] —-> R be given by g(x) = f(d‘x).
Then t Z 0, and hence g is expandable on [d“oz,d“fi]. Thus, by Theorem 6.1, 9
assumes on [d‘toz, d"fl] every intermediate value between g(d“oz) and g(d“fl). Now
let S be an intermediate value between f (a) and f (,8); then S is an intermediate

value between g(d“a) and g(d"fl). Hence there exists 7 E [d"a,d“fl] such that

9(7) = S. Let 77 = d‘y. Then 77 E [07,6] and f(77) = g(d"n) = 9(7) = 5',

Corollary 6.6 Let a < b in R, let a < B in [a, b] be given and let f : [a, b] —» R
be expandable on [a, b]. Then f assumes an [a, 6] every intermediate value between

f (a) and f (6)-

185

Proof. Let t = A(b— a) and let g : [d“a,d“b] —-> R be given by g(x) = f (d‘x).
Then g is expandable on [d“a,d“b] by definition, where 0 < d“b — d"a ~ 1. By
Corollary 6.5, 9 assumes on [d“a,d"fl] every intermediate value between g (d‘ta)

and g (d“fl). It follows that f assumes on [a, 6] every intermediate value between

f (a) and f (B).
6.2.2 Maximum Theorem and Mimimum Theorem

We start this section by stating the following theorem whose proof follows directly
from that of Theorem 5.20.

Theorem 6.2 Let a,b E R be such that 0 < b — a ~ 1, and let f : [a, b] ——> R be
expandable on [a, b] with i( f) = 0. Then f is infinitely often differentiable on [a, b],
and for all m E Z+, we have that f (m) is expandable on [a, b]. Moreover, if f is given

locally around x0 E [a, b] by f (x) = 23.10% (x0) (x — x0)", then f(ml is given by

00

7"") (x) = gmor) = Z n(n—1)~-(n— m+1)an(mo)(x- so)”:

n=m

In particular, we have that am (x0) = f(m) (x0) /m! for all m = 0, 1, 2, . . ..

Corollary 6.7 Let a < b in R be given, and let f : [a, b] —-> R be expandable on
[a, b]. Then f is infinitely often differentiable on [a, b], and for all m E Z+, we have
that f (m) is expandable on Ia, b]. Moreover, if f is given locally around x0 E [a, b] by

f (x) = 23100,: (330) (SE - mo)". then f‘m’ is given by

f((m x()= Z n (n — 1) (n m + 1) an (x0) (x -- x0)"_m.

n=m

In particular, we have that am (x0) = f(m) (x0) /m! for all m = 0, 1, 2, . . ..

The following theorem is again a generalization of the maximum theorem for

normal functions [5]; the key step in the proof is to apply the intermediate value the—

186

orem, Theorem 6.1, to the derivative function which is itself an expandable function

by Theorem 6.2.

Theorem 6.3 Let a,b E R be such that 0 < b- a ~ 1, and let f : [a,b] ——> R be

expandable on [a, b]. Then f assumes a maximum on [a, b].

Proof. Without loss of generality, we may assume that i( f) = 0 and that fR is not
constant on [03,63] 0 R, where an = %(a) and b}; = %(b). Since f3 is continuous
on [ambn] F) R, f}; assumes a maximum MR on [ambn] n R. Since f]; is analytic
on [am b3] 0 R, there are only finitely many points X1, . . . , X), in [am b3] 0 R where
f}; assumes its maximum M R. We look for a maximum of f in the infinitely small
neighborhoods of the Xj’s. So let 7' E {1,...,k} be given. Assume X,- E (aR,bR).
Then f}; (X j) = 0 and there exists 61 > 0 in R such that

fII(X) > OforX E (Xj—61,Xj)nRand

fHX) < Ofor X e (X,,X,-+61)nR.

Using Theorem 6.2 and the fact that f is expandable at X j, there exists 6 S 61 in R
such that 0 S IhI < 6 =>

M(X)

——h"‘1.
_ 1)7

f(X,-+h)== —ZI———(n:2(!X handf( (X-)-—-+h —Z(—::——

n=0

Let

m=min{nEZ+: A(X(f("+1) -))=0}.
Using the intermediate value and its proof, and using Remark 6.3 and its proof, all
applied to f’, we obtain at least one and at most (m — 1) roots of f’ that are infinitely
close to X j, and f’ changes sign from positive to negative in going from the left to the
right of at least one of the roots. Thus we obtain at least one and at most (m — 1)

local maxima of f in the infinitely small neighborhood of X j. Let

Mj=max{f(X,-+h) : Ih|<<1}.

 

187

Similarly we show that f has a maximum in the infinitely small neighborhood of a if
an E {X1, . . . ,Xk} and that f has a maximum in the infinitely small neighborhood
Ofbibe E {X1,...,Xk}. Let

M=max{M_,-:1SjSk}.

We Show that M = max{f (x) : x E [a,b]}. So let x E [a, b] be given. Suppose x is

finitely away from X,- for all j E {1, . . .,k}. Then
f(l‘) — M = (f (a?) - fR(§R($))) + (hem-Tl) - ER(1"1))+(§R(1"1) " M)-

Since ER (2:) E {X1, . . . , Xk} , we have that f}; (9? (x)) — %(M) is negative and finite in
absolute value. Since If (x) - f}; (R (x))l << 1 and since |9‘i(M) — M] < 1, we obtain
that f(x) — M < 0; that is f(x) < M.

Now suppose a: is infinitely close to one of the X ,- ’s, say X ,0. Then f (x) S MJ-o S
M. Thus f (x) S M for all x E [a, b]. Moreover, M is assumed on [a, b]. Hence

M=max{f(x) :xE [a,b]}.

Corollary 6.8 Let a,b E ’R. be such that O < b — a ~ 1, let a < 6 be given in [a, b]

and let f : [a, b] —-> ’R be expandable on [a, b]. Then f assumes a maximum on [01,6].

Corollary 6.9 Let a < b in R be given, let 01,6 E [a, b] be such that a < 6 and let

f : [a, b] -> ’R be expandable on [a, b]. Then f assumes a maximum on [a, 6].

Proof. Let t = /\(b - a) and let 9 : [d“a,d“b] —-> ’R. be given by g(y) = f (d‘y).
Then 9 is expandable on [d“a,d“b]. Thus there exists yl E [d“a,d“fl] such that
g (y) S g (yl) for all y E [d"a, d“B] . Let x1 = d‘yl, and let x E [01,6] be given. Then
d“x E [d“a,d"fl] . Thus

f (x) = 9 (Wm) S 9 (311) = g (d“a:1) = f (361).

Hence f(x) S f (231) for all x E [a,fl].

 

188

Corollary 6.10 Let a < b in R be given, let 01,6 E [a, b] be such that (1 < 6 and let

f : [a, b] —* R be expandable on [a, b]. Then f assumes a minimum on [a, 6].

Corollary 6.11 Let a < b in R be given, let (1,6 E [a, b] be such that a < 6 and let

f : [a, b] —> R be expandable on [a, b]. Then there exist m, M E R such that
f([aiflll = [m’M] °

Proof. By Corollary 6.9 and Corollary 6.10, there exist x1,x2 E [(1,6] such that
f(x1)S f(x) S f(xg) for all x E[(1,6]. Let m = f(xl) and M = f($2). By Corollary
6.6, for each 3; E [m, M], there exists an E [231,272] C [(1, 6] such that f (x) = y. Thus,
f([aflll = [WM]-

6.2.3 Rolle’s Theorem and the Mean Value Theorem

In this section, we prove Rolle’s theorem and the mean value theorem for expandable

functions, which will lead the way to an integration theory in Section 6.3.

Theorem 6.4 (Rolle’s Theorem) Let a < b in R be given, let a, 6 E [a, b] be such
that a < 6 and let f : [a, b] —* R be expandable. Suppose f (a) = f (6). Then there

exists c E ((1,6) such that f’ (c) = 0.

Proof. If f(x) = f(a) for all x E [(1, 6], then f’ (x) = 0 for all x E ((1,6) and we
are done. So we may assume that f is not constant on [a, 6]. Then f has either a
maximum or a minimum at some c E ((1,6). Using Corollary 6.7 and Lemma 5.9, we
obtain that f is topologically differentiable at c. Using Theorem 5.7 and Corollary
5.6, we finally obtain that f’ (c) = 0.

Like the intermediate value theorem and the maximum theorem, the following
mean value theorem is a generalization of the corresponding result for the normal

functions [5].

I:

 

189

Theorem 6.5 (Mean Value Theorem) Let a < b in R be given, let (1,6 E [a, b]
be such that a < 6 and let f : [a, b] —> R be expandable on [a, b]. Then there exists
c E ((1, 6) such that

’C _f(:8)_f(a)
f()— 3% .

 

Proof. Let F : [a, b] —» R be given by

Then F is expandable on [a, b]. Moreover,
P(a) = ("(6) = 0.

Thus, by Theorem 6.4, there exists c E ((1, 6) such that F’ (c) = 0; that is

_ f(fi) -f(a)

o = F'(c) = f’(c) 5w ,

which finishes the proof of the theorem.

As a direct consequence of the Mean Value Theorem, we obtain the following

important result.

Corollary 6.12 Let a < b in R be given, and let f : [a, b] —» R be expandable on

[a, b]. Then the following are true.

(i) If f’ (x) 79 0 for all x E (a, b) then either f’ (x) > 0 for all x E (a, b) and f
is strictly increasing on [a, b], or f’ (x) < O for all x E (a, b) and f is strictly

decreasing on [a, b].

(ii) If f’ (x) = O for all x E (a, b), then f is constant on [a, b].

Proof.

 

190

(i) Suppose f’ (x) 79 0 for all x E (a, b). Then applying the intermediate value
theorem to f’, we infer that either f’ (x) > 0 for all x E (a, b) or f’ (x) < O for
all x E (a, b). First assume that f’ (x) > 0 for all x E (a, b) and let x,y E [a, b]
be such that y > x. By Theorem 6.5, there exists c E (x, y) C (a, b) such that

f(y)-f(x)
y-x '

 

f' (C) =

Since c E (a, b) , we have that f’ (c) > 0; and hence f (y) > f (x). Thus, f is

strictly increasing on [a, b].

Now assume that f’ (x) < 0 for all x E (a, b) and let x,y E [a, b] be such that

y > x. Then there exists c E (x, y) C (a, b) such that

=f(y)-f(x).

f’(c) 3H,

Since c E (a, b) , we have that f’ (c) < 0; and hence f (y) < f (x). Thus, f is

strictly decreasing on [a, b].

(ii) Suppose f’ (x) = O for all x E (a, b), and let y E [a,b] be given. There exists

c E (a, y) C (a,b) such that

 

, _f(y)-f(a)
f(c)— y-.. .

Since c E (a, b) , we have that f’ (c) = 0; and hence f (y) = f (a) . Thus f (y) =
f(a) for all y E [a, b] .

6.3 Integration

In this section, we develop an integration theory on the class of expandable fimctions,
which is a generalization of the integration theory developed in [5] for the normal

functions.

 

191

Definition 6.5 Let a < b in R be given, and let f : [a, b] -» R be expandable on
[a, b]. Then a function F : [a, b] —i R is said to be an expandable primitive of f on
[a, b] if and only if F is expandable on [a, b] and F’ (x) = f (x) for all x E [a, b].

Lemma 6.4 Let a,b E R be given such that 0 < b — a ~ 1, and let f : [a, b] —+ R be

expandable on [a, b]. Then f has an expandable primitive F on [a, b].

Proof. Using the proof of Lemma 6.3, there exists m E Z + and there exist x1, . . . , xm E F
[a, b] such that

[a, bl C U311 (932' — 5(leixj + 5051)),

where for all j E {1, . . . ,m}, 6(xj) is a real domain of expansion of f around xj. For

 

all j E {1, . . . ,m}, there exists a regular sequence (afl(x,-)) in R such that

f(x) = i an(x,-)(x — xj)" for all x E (x,- — 6($j),$j + 6(xj)).

n=0
Define F : [a, b] -+ R by

(X)

F(x) = Z (:"ixfi!(x — xj)"+1 for all x E (x, — 6(xj),x,- + (S(x,-D;

 

n=0
for all j E {1, . . . ,m}. Using Theorem 5.21, we obtain that F is expandable on [a, b].

Using Theorem 6.2, we obtain that
F'(x) = f(x) for all x E [a, b].

Hence F is an expandable primitive of f on [a, b].

Corollary 6.13 Let a < b in R be given, and let f : [a, b] —-> R be expandable on

[a, b]. Then f has an expandable primitive F on [a, b].

Lemma 6.5 Let a < b in R be given, and let f : [a, b] —» R be expandable on [a, b].

Let F1 and F2 be two expandable primitives of f on [a, b]. Then there exists a constant

c E R such that F2 (x) = F1 (x) + c for all x E [a, b].

192

Proof. Let F : [a, b] -» R be given by F (x) = F2 (x) — F1 (x). Then F is expandable
on [a, b] and F’ (x) = F; (x) —F1’ (x) = f (x) —f (x) = 0 for all x E [a, b]. By Corollary

6.12, F is constant on [a,b].

Corollary 6.14 Let a < b in R be given, let (1, 6 E [a, b] be given and let f : [a, b] —>

R be expandable on [a, b]. Let F1 and F; be two expandable primitives of f on [a, b].
Then F2 (6) — F2 (or) = F1(6) - F1(a).

Definition 6.6 Let a < b in R be given, let (1,6 E [a, b] be given and let f : [a, b] —+ R
be expandable on [a, b]. We define the integral of f from a to 6, denoted by ff f, as

follows: Let F : [a, b] —-> R be an expandable primitive of f on [a, b], which exists by

 

Corollary 6.13 and let ll!
6
Lf=F(fl)-F(a)-

Remark 6.4 By Corollary 6.14, the integral in Definition 6.6 is independent of the
choice of the expandable primitive function F; it depends only on f, a, and 6, and

hence it is well defined.

Theorem 6.6 Let a < b in R be given, let (1,7,6 E [a, b] be given, let f,g : [a, b] —> R

be expandable on [a,b] and let It E R be given. Then

fa...) = fin/jg. ....

[ff [Hffl

Proof. Let F ,G : [a, b] —> R be expandable primitives of f and g on [a, b], re-

spectively. Then, using Lemma 6.1, we obtain that F + KG is expandable on [a, b].

Moreover, using Theorem 5.13, we obtain that

(F + KG), (x) = F'(x) + nG’(x) = f(x) + ng(x) = (f + my) (x) for all x E [a, b].

193

Thus, F + RC is an expandable primitive of f + reg on [a, b]; and hence

f(f + ..g) = (F + new) — (F + ncxa)

= F(g) — F(a) + n (0(6) - GW)
3 fl
= Lf+nfa 9-

Since F is an expandable primitive of f on [a, b], we also have that

B
l. f = F(fl)-F(a)

= (F(V) - F(a)) + (F(fi) - F(x))

= [f+[ff.

Chapter 7

Computer Functions

In this final chapter, we present one of the applications of the non-Archimedean field
R, namely the computation of derivatives of real functions that can be represented

on a computer; see also [38, 39, 40, 42].

7. 1 Introduction

The general question of efficient differentiation is at the core of many parts of the work
on perturbation and aberration theories relevant in Physics and Engineering; for an
overview, see for example [11]. In this case, derivatives of highly complicated functions
have to be computed to high orders. However, even when the derivative of the function
is known to exist at the given point, numerical methods fail to give an accurate value
of the derivative; the error increases with the order, and for orders greater than three,
the errors often become too large for the results to be practically useful. On the other
hand, while formula manipulators like Mathematica are successful in finding low-order
derivatives of simple functions, they fail for high-order derivatives of very complicated

functions. Consider, for example, the function

 

- 3 3+cos(sin(ln|l+x|))
3111 x + 23 + 1 + . sin cos tan ex 2:
< ) exp(mtmhw.....2..§...e..::.i;i.>>)) (, 1,

2 + sin (sinh (cos (tan'1 (1n (exp(x) + 2:2 + 3)))))

 

g(x) =

194

 

Using the R numbers implemented in COSY INFINITY [8, 12], we find g(")(0)

for O S n S 19. These numbers are listed in Table 7.1; we note that, for 0 S n S 19,

195

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Order n g(")(0) CPU Time
0 1.004845319007115 1.820 msec
1 0.4601438089634254 2.070 msec
2 —5.266097568233224 3.180 msec
3 -—52.82163351991485 4.830 msec
4 —108.4682847837855 7.700 msec
5 16451.44286410806 11.640 msec
6 541334.9970224757 18.050 msec
7 7948641.189364974 26.590 msec
8 —144969388.2104904 37.860 msec
9 —15395959663.01733 52.470 msec
10 -618406836695.3634 72.330 msec
11 —11790314615610.74 97.610 msec
12 403355397865406.1 128.760 msec
13 0.5510652659782951 x 1017 168.140 msec
14 0.3272787402678642 x 1019 217.510 msec
15 0.1142716430145745 x 1021 273.930 msec
16 —0.6443788542310285 x 1021 344.880 msec
17 -0.5044562355111304 x 1024 423.400 msec
18 —0.5025105824599693 x 10” 520.390 msec
19 —0.3158910204361999 xi” 621.160 msec

 

Table 7.1: g(")(0), 0 S n S 19, computed with R calculus

we list the CPU time needed to obtain all derivatives of g at 0 up to order n and
not just g(")(0). For comparison purposes, we give in Table 7.2 the flmction value
and the first six derivatives computed with Mathematica. Note that the respective
values listed in Table 7.1 and Table 7 .2 agree. However, Mathematica used a much
longer CPU time to compute the first six derivatives, and it failed to find the seventh
derivative as it ran out of memory. We also list in Table 7.3 the first ten derivatives

of g at 0 computed numerically using the numerical differentiation formulas

900(0) = We)“ (iv-1W ( g? )g 0420) , Ax = 10-16/<"+1>,

i=0

 

 

196

for 1 S n S 10, together with the corresponding relative errors obtained by comparing

the numerical values with the respective exact values computed with R calculus.

 

 

 

 

 

 

 

 

 

 

 

Order n gl")(0) CPU Time
0 1.004845319007116 0.11 sec
1 0.4601438089634254 0.17 sec
2 -5.266097568233221 0.47 sec
3 —52.82163351991483 2.57 sec
4 -108.4682847837854 14.74 sec
5 16451.44286410805 77.50 sec
6 541334.9970224752 693.65 sec

 

 

Table 7.2: g(")(0), O S n S 6, computed with Mathematica

 

 

 

 

 

 

 

 

 

 

 

Order n 900(0) Relative Error
1 0.4601437841866840 54 X 10"9
2 —5.266346392944456 47 X 10—6
3 —52.83767867680922 30 X 10-5
4 -87.27214664649106 0.20
5 19478.29555909866 0.18
6 633008.9156614641 0.17
7 —12378052.73279768 2.6
8 —1282816703.632099 7.8
9 83617811421.48561 6.4
10 91619495958355.24 149

 

 

 

 

 

Table 7.3: g(")(0), 1 S n S 10, computed numerically

Furthermore, numerical methods and formula manipulators fail to find the deriva-
tives of certain functions at given points even though the functions are differentiable

at the respective points. For example, the functions

——*21—11-° $-42 -g(x) 11274 0

0 if x = O ’ (72)

91(3) 2 [$[5/2 . g(x) and g2(x) = {

where g(x) is the function given in Equation (7.1) above, are both differentiable at
0; but the attempt to compute their derivatives using formula manipulators fails.

This is not specific to 91 and 92, and is generally connected to the occurrence of

197

nondifferentiable parts that do not affect the differentiability of the end result, of
which case g1 is an example, as well as the occurrence of branch points in coding as

in IF-ELSE structures, of which case g2 is an example.

One of the applications of the non-Archimedean field R deals with many of the
general problems connected to computational differentiation [38, 39, 40]. Using the
calculus on R, we formulate a necessary and suflicient condition for the derivatives
of functions from R into R representable on a computer to exist, and show how to

find these derivatives whenever they exist.

7 .2 Computer Functions of One Variable

At the machine level, a function f : R —+ R is characterized by what it does to the
original set of memory locations. So f induces a function F( f) : R'" —> R'", where
m is the number of memory locations affected in the process of computing f. We
note here that, without compiler optimization, F( f ) is unique up to flipping of the
memory locations; on the other hand, with compiler optimization, F( f) is unique in
the subspace describing the true variables. Moreover, at the machine level, any code
constitutes solely of intrinsic functions, arithmetic operations and branches. In the
following, we formally define the machine level representations of intrinsic functions,

the Heaviside function, and the arithmetic operations.

Definition 7 .1 Let I: {H , sin, cos, tan, exp, . . .} be the set consisting of the Heavi-
side function H and all the intrinsic functions on a computer, which for the sake of

convenience are assumed to include the reciprocal function; and let 0: {+, }.

Definition 7.2 For f e 1', define F1), , : Rm —» R’" by

—0

E,k,f(mli$2i ' ' ° ixm) = ($1, ' ° ° ixk—1,f(xi))xk+1" ° ° ixm);
k

is...“

 

198

so the kth memory location is replaced by f (x,) Then 1?ng is the machine level

representation of f. For <8) E 0, define Fig-kg : R'" —+ R“ by

4

E,j,k,®($1, 1'2, . . . , (rm) 2 ($1, . . . ,SL'k_1, IL',‘ ® 183', $k+1, . . . ,xm),
W
I:

so the kth memory location is replaced by 2:.- ® xj. Then F‘Mfifl is the machine level

representation of 69. Finally, let

f = {Flu ; f e I} u {FL-m = 99 E 0}-

Definition 7.3 A function f : R —i R is called a computer function if and only
if it can be obtained from intrinsic functions and the Heaviside function through a
finite number of arithmetic operations and compositions. In this case, there are some
F1,F2,...,FN E .7: such that F(f) = FN o FN_1 o 0 F2 0 F1, and we call F(f):

R” -+ R'", already mentioned above, the machine level representation of f.

Obviously, the so defined class of computer functions in a formal way describes all
those functions that can be evaluated on a computer. Since we will be studying only
computer functions, it will be useful to define the domain Dc of computer numbers

as the subset of the real numbers reprasentable on a computer.

We recall the following result, Corollary 4.10, which allows us to extend all intrinsic

functions given by power series to R.

 

Theorem 7 .1 (Power Series with Purely Real Coefficients) Let Em anX", an E

n=0
R, be a power series with classical radius of convergence equal to 77. Let x E R, and
let A,.(x) = 2&0 aixi E R. Then, for |x| < 17 and [x] 56 17, the sequence (An(x))
converges absolutely weakly. We define the limit to be the continuation of the power

series on R.

199

Remark 7.1 The continuation H of the real Heaviside function H is defined for all

xERby

- _ 1 ifoO
H(x)"[0 ifx<0'

The functions {75 and l/x are continued to R via the existence of roots and

multiplicative inverses on R (see Section 3.1).

Definition 7.4 Let f E I, let D be the domain of definition of f in R, let x0 E D,
and let 3 E R. Then we say that f is extendable to x0 + 3 if and only if x0 + 3 belongs

to the domain of definition of f, the continuation of f to R, where f is given by
Theorem 7.1 and Remark 7.1.

Let f1, f2 E I with domains of definition D1 and 02 in R respectively, let x0 E
D1 (‘1 D2, let 3 E R, and let 8) E {+,'}. Then we say that f2 (8 fl is extendable to

x0 + 8 if and only if f1 and f2 are both extendable to x0 + 3.

Let f1, f2 E I with domains of definition D1 and D2 in R respectively, let x0 E DI
be such that f1(xo) E D2, and let s E R. Then we say that f2 0 fl is extendable to

x0 + 3 if and only if fl is extendable to x0 + s and f2 extendable to f1(xo + 3).

Finally, let f be a real computer function, let D be the domain of definition of
f in R, let x0 E D, and let 3 E R; then f is obtained in finitely many steps from
functions in I via compositions and arithmetic operations. We define extendability

of f to x0 + s inductively.

We have the following rasult about the local form of computer functions, which

will prove useful in studying the differentiability of computer functions.

Theorem 7.2 Let f be a real computer function with domain of definition D, and

let x0 E D be such that f is extendable to x0 i d. Then there exists a > 0 in R such

 

200

that, for 0 < x < (7,

f(xo i x): A€(x) )+ :xq‘ Af( (x,) (7.3)

i=1
where A,-i(x),0 Si Si i, is a power series in x with a radius of convergence no
smaller than a, Af(O) aé 0 fori = 1,...,ii, and the qu’s are nonzero rational

numbers that are not positive integers.

Remark 7 .2 Noninteger rational powers may appear in Equation (7.3) as a result

of the root function.

Proof. The statement of the theorem can easily be verified for each f E I.

Let f1 and f2 be two computer functions with domains of definition D1 and D2 in
R, respectively. Let x0 E D1 0 D2, let f1 and f2 be both extendable to x0 i d, and let
f1 and f2 satisfy Equation (7.3) around x0. For <8) E {+, -}, let F8, = f2 ® f1. Thus
we have that

ii:
f1(x0 :l: x) = Affix) + quiiAflx) for x E (0,01),

i=1

ji:
f2(xo :l: x) = 30*(23) + Zx‘fob) for x E (0,02),
j=1

where 01 and (72 are both positive real numbers; Af(x),0 S i S i*, and Bflx), 0 S

j S j i, are power series in x with radii of convergence no smaller than a = min{ol, (12};

Af(O) 7e 0 for i E {1,...,ii} and Bf(0) 76 O for j E {1,...,ji}; and the qf’s and
the t-i’s are nonzero rational numbers that are not positive integers. As a reminder,

we note that 01, (72, the Ai’s, the Bf’s, theq ’s, and the tf’s depend on re.

For 0 < x < a, we have that

F®(xo :l: x): f2(xo :i: 2:) <83 f1(xo :t x) =(f: x‘1i Af(x) )) <8) (:5: x‘fBJi(x)) , (7.4)
j=0

i=0

 

[i

201

where q? = t3 = 0. It is easy to check that, for <8) = + or (8 = -, the result in

Equation (7.4) is an exprefision of the form of Equation (7.3).

Now let f1 and f2 be two computer functions with domains of definition D1 and Dz
in R, respectively. Let x0 E D1, let f1 be extendable to x0 :1: (1, let f2 be extendable to
f1(xo:l:d), and let f1 and f2 satisfy Equation (7.3) around x0 and f1(xo), respectively.
Let Fo = f2 0 f1. Thus we have that

f1(xo :l: x) = Affix) + ixqiiAflx) for x E (0,01),

i=1

f2(f1(xo)=ty) = Bi(v)+ y Bf(v)f0rv€(0,02).

where (71 and 02 are positive real numbers; Af(x), 0 S i S ii and Bf(y), O S j S j*,
are power series in x and y with radii of convergence no smaller than a = min{01, 02};
Af(O) 31$ 0 for i E {1,...,ii} and Bf(0) aé 0 forj E {1,...,j‘h}; and the qjt’s and
the tf’s are nonzero rational numbers that are not positive integers. Without loss of
generality, we may assume that at least one of the series Bf (y) is infinite. It follows,
since f2 is extendable to f1(xo i d), that the qf’s are all positive and that 143(0) =
f1(x0). Let A§0(x) = .43: (x) -— 1435(0) = Aflx) — f1(xo). Then A§0(x) has no constant
term, and we have, for 0 < x < (71, that f1(xo:l:x) = f1(x0)+Af]:o(x)+Z':1 xquflx).
Since A§0(x) has no constant term and the qf’s are all positive, there exists 0’ E R, 0 <
0' S 01, such that [A30(x) +21; x‘lii Af(x)| < (72 and Af(,(x) +21; quhA,‘-t(x) has the
same sign for all x satisfying 0 < x < a. To prove the last statement, note that since
f(x) = A§0(x) + 2:1 xqiiAflx) is continuous at 0, there exists 61 E R, 0 < 61 S 01,
such that 0 < x < 61 => |g*(x) — gi(O)I = |A0i0(x) + 22:1 $9?Af(x)| < (72. Now let
(1*in be the leading term of gi (x). Write g2t (x) = ozix‘li (1 + git(x)), where git (x)
is continuous at 0 and git(0) = 0. Hence there exists 62 E R, 0 < 62 S 01, such that

0 < x < (52 => |gf(x)| < 1/2 => 1+ gflx) > 0 => g*(x) has the same sign as 01*. Let

 

202

0' = min{(51,62}. Then 0 < a S (71, and 0 < x < a => |A€0(x) + 21:1 xqiiA,-i(x)| < (72
and Affo(x) + 27:1 xquflx) has the same sign as (1*. Thus, for O < x < a, we have

that

Fo($0 i 93) = f2 (f1($0 :l: 17)) = f2 (f1(330) + 1460604“ :xquf:(x))
= E0 (480(4) + 2424666))

J

+ Z{ A00(x) + 2in Af(x)

=1

 

 

E-(Aoo(x) + qui Af(x)) },

i=1
where Ej,0 S j S J, are power series; Ej(0) 79 0 for 1 S j S J; and the sj’s are
nonzero rational numbers that are not positive integers.

Note that for 1 Sj S J,

81'
:l:

 

: la 83' xsjqi (1+gl(x ))BJ'

i=1:
+ 2 x9? Af(x)
i=1

 

 

:i:

x 5.491 (x)),

 

[(1

where gl i'(x) is of the form of Equation (7. 3), gli(0) = O, Igli(x)| < 1/2, and Sj(gf‘ (x)) =
(1 + gli(x))8’ is a power series in g1DE (x) Thus, it suffices to show that a power series
of an expression of the form of Equation (7.3), in which the qfh’s are all positive and

in which A§(0 )- — , Oyields an expression of the same form.

So let S ( )= £m_0 amym be a power series with positive radius of convergence 17.

Then, for x sufficiently small,

i=1 m=0

S (Aflx) + :xquf(x))= i am (A0 (x) + qui Af( x()) . (7.5)

For each i E {1,... ,i,*} write q,- — —m:h/n,- , where m? and nit are positive and

relatively prime. Expanding the powers in Equation (7.5), the only exponents of x

 

203

that may occur are of the form k + s, where k is a positive integer and

.4 at
..T:{%,...,(n:—1)g4/.=1,...,.4},

a finite set. For each m let Sm(x) = am (Aflx) + 22:1 xqiiAf(x))m. Then Sm is an
infinite series

Sm(x) = i umn(x), (7.6)

n=0

where umn(x) is of the form amnxkfl with amn E R, k a positive integer, and s E T.
Let 171 be the radius of convergence of Aoi(x) + 22:1 xqfii A? (x), and let 0 < x < 171/2
be such that lAfo) + 21:1 x93: Af(x)] < 71/ 2. Then for each m, the sum in Equation
(7.6) converges absolutely; so we can rearrange the terms in Sm. Moreover, the double

sum Ema-0 3:0 umn(x) converges; so (see for example [31], pages 205-208) we obtain

that

1* oo oo oo 00
S (Aflx) + 2396Af(x)) = Zozoumfix) = 20 2011"",(x).
Thus rearranging and regrouping the terms in Equation (7.5), we obtain an expression
of the form Cfflx) + 2;; x'th;h(x), where C:(x),0 S p S p*, are power series,
03(0) 75 0 for 1 S p S pi, p:h is finite, and the rf’s are nonzero rational numbers
which are not positive integers. Hence S (Aflx) + 22:1 x‘Iii Iii-h(x)) is of the form of
Equation (7.3). It follows that Fo(xo :l: x) in Equation (7.5) is itself of the form of

Equation (7 .3).
Now let f be a real computer function with domain of definition D, and let :60 E D
be such that f is extendable to x0 :1: d. Then f is obtained in finitely many steps

from functions in I via compositions and arithmetic operations. Using induction, we

obtain the result immediately from the above.

Since the family of computer functions is closed under differentiation to any order

71, Theorem 7.2 holds for derivatives of computer functions as well.

 

204

Definition 7 .5 (Continuation of Real Computer Functions) Let f be a real computer
function with domain of definition D and let x0 E D be such that f is extendable to
x0 :t d. Then f is given around x0 by a finite combination of roots and power series.
Since roots and power series have already been extended to R, f is extended to R

around x0 in a natural way similar to that of the extension of power series from R

4
to C. That is, if f(xo :l: x) = Af(x) + 2:; xqitA$(x) for O < x < a, then we have
-:i:

for the continued function f that f(xo :l: x) = A?)t (x) + 2:: xquflx) for all x E R

satisfying 0 < x < (7 and x at (7.

Theorem 7.3 Let f be a computer function that is diflerentiable at the point x0 E R
and extendable to x0 :l: d. Then the continued function f is topologically difl'erentiable

at x0, and the derivatives of f and f at x0 agree.

Proof. Since f is differentiable at x0, there exists a > O in R such that, for x E R
and 0 < x < (7, f(xo j: x) = f(xo) :l: f’(xo)x + 23:2 (fixi + 231:1 xquflx); where
qf, . . . ,qft are noninteger rational numbers greater than 1, and A3, Ait, . . . ,Aft are
power series in x. Thus we have for the continued function f that f (x0 1 x) =
f(xo) :1: f’(xo)x + 2222 afx‘ + 2}.: xquflx) for all x E R satisfying 0 < x < o and

x at: a. Let

4_ min{qf;lSjSJ*} if{qf;1gng*}¢0
q _ 0° if{q,-*;1_<.15J=*=}=0’

let q = min(q+,q‘), and let k = min{1,q — 1}. Then 0 < k S 1. We show that

the continued function f is topologically differentiable at x0, with derivative f’ (x0) =
f '(xol-

Let x E R satisfy 0 < x < (7 and x at (7. Then we have that

f(xoixl ‘ f(mo)

00 J3:
i— .i—
(:tx) i2 (ifx l :l: E x": 1Af(x) .

i=2 j=l

 

- f '(130)

 

 

 

205

Let c > 0 be given in R. As a first case, assume 6 is not infinitely small, and let 5, =

1 :1:
as y —> 0+,y E R, is equal to zero, there exists 6 E R, 0 < 6 < 0/2, such that

f($0iy)—f($0) _
(iv)

Now let x E R be such that 0 < x < 6, and let xr = .‘R(x). If x, = 0, then x

i-

62(6), the real part of 6. Since the real limit of lriz 22:2 (1ihyi’1 i 2;: ya,

 

f '(30)

 

 

< :25 whenever y E R and 0 < y < 26.

is infinitely small. Thus |{f(xo :l: x) — f (xo)} / (:lzx) — f’ (x0)| is infinitely small, and

hence smaller than c. If x, 74 0, then 0 < x, < 26. Therefore,

litre 2*: x) - f(370)
(ix)

f(xO i (t,.) - f(xO)
(:i:x,.)

6r
< —.
2

 

 

— f'($0)] =0

 

- f'($o)

 

Hence ]{f(xo + x) — f(x0)}/x — f’(x0)[ < 6 whenever 0 < [x] < 6.
As a second case, assume 6 is infinitely small. Let

4_ min{iz2:a,*;40} if{422:a3=740}¢0
m — oo if{z‘22:a,*;é0}=0'

If m:t = 00, let 01:: = 0. With the convention 1/0 = 00, let

1 , l/k _ l/k l/k _ 1/1:
6 = 5m [(e/IANOM) .(e/IA1(0)I) , (4141.1) .(e/lam-I) }
Then 6 > 0, and ifO < [x] < 6 then |{f(xo +x) — f(xo)}/x — f’(xo)| < 6. Thus f is
topologically differentiable at 2:0, and f’(x0) = f’ (x0).

In the rest of this chapter we will use f instead of f to represent the continuation

of a real computer function f.

7 .3 Computation of Derivatives

In this section, we develop a criterion that will allow us not only to check the continuity
and the differentiability of a real computer function f at a point x0, but also to obtain

all existing derivatives of f at x0.

 

206

Lemma 7 .1 Let f be a computer function. Then f is defined at x0 E Dc if and only

if f (x0) can be evaluated on a computer.

This lemma of course hinges on a careful implementation of the intrinsic functions
and operations, in particular in the sense that they should be executable for any
floating point number in the domain of definition that produces a result within the

range of allowed floating point numbers.

Lemma 7.2 Let f be a computer function, let D be the domain of definition of f in
R, let x0 E D 0 DC, and let 3 E R. Then f is extendable to x0 + 8 if and only if

f (x0 + s) can be evaluated on the computer.

Lemma 7.3 Let f be a computer function, and let x0 be such that f is defined at x0

and extendable to x0 :1: d. Then f is continuous at x0 if and only if

f(on - d) =0 f(xo) =0 f(xo + d)-

Proof. Since f is a computer function, defined at x0 and extendable to x0 :I: d, we
have that
J. J!
f(xo + x) = Ao(x) + JEx‘b'Ajw) and f(xo — x) = Bo(x) + gx‘jBJ-(x)

for 0 < x < (7, where a is a positive real number; where the Aj’s and the Bj’s are
power series in x, where Aj(0) 51$ 0 for 1 S j S J, and Bj(0) 91E 0 for 1 S j S J1; and
where the qj’s and the t,- ’s are nonzero rational numbers that are not positive integers.
Let Ao(x) = 2:0 (1.x" and Bo(x) = 2:0 ,x‘. Then f is continuous at x0 if and only
ifq,- > O for allj E {1,...,Jr}, tj > 0 for allj E {1,...,J1}, and do = 60 = f(xo);
that is, if and only if f (x0 + d) =0 f (x0) =0 f (x0 — d).

 

207

Theorem 7 .4 Let f be a computer function that is continuous at x0 and extendable
to x0 :1: d. Then f is diflerentiable at x0 if and only if (f (x0 + d) -- f (xo)) /d and
(f (x0) — f (x0 — (1)) /d are both at most finite in absolute value, and their real parts

agree. In this case,

“1170 +60 - f(930)
d

f(330) - f(xo "- d)
d .

 

=0 f'($0) =0

 

If f is differentiable at x0 and extendable to xoid, then f is twice differentiable at x0 if
GM only if (f(1130+2d)-”(5504-60‘l‘fC’BOD/d2 and (f(flvo)=2f(fl70"(l)“l'f(3?30—2d))/d2

are both at most finite in absolute value, and their real parts agree. In this case

f(xo+2d) — 2f<xo+d) +f(xo) = M )_ f(xo) — 2m. — 4) +f($o-2d)
d2 0 $0 —'0 d2 .

In general, if f is (n — 1) times differentiable at x0 and extendable to x0 i d, then

f is n times differentiable at x0 if and only if d’“ ( ;‘=0(—1)"‘j ( 3L ) f (x0 + 311))

and d"" ( 31:0(_1)j ( 3 ) f (x0 — jd)) are both at most finite in absolute value, and

their real parts agree. In this case

51—" (g)(‘1ln—j ( g ) f(5'30 +160) =0 f(n)($0)

:0 d‘" (g)(-16 ( 2’ ) f(l‘o — jdl) -

Proof. Since f is continuous at x0, we have that

00 J,-
f(£L‘o + 1‘) = f(ato) + 2041f. + ZIIIQjAj(.’E)
i=1 j=1

00 J;
f(IL‘o — (I?) = f(xo) + 264119.“? 2: xtJ'Bj(x) (7.7)

i=1
for 0 < x < o, where a is a positive real number, where the Aj’s and the Bj’s are

power series in x that do not vanish at x = 0, and where the qj’s and the tj’s are

 

208

noninteger positive rational numbers. Observe that f is n times differentiable at x0
if and only if
q,->nfor1SjSJ,.,tj>nfor1SjSJz,andaj=(—1)j6jforlSan.
(7.8)

Assume f is differentiable at x0. Then, using Equation (7.8), we have that
q, > 1 for alle {1,...,Jr}, t,- >1 for allj E {1,...,J1}, and (11 = —61=f’(x0).

Hence,

f(x“ + 6131— f(xo) = iafl‘l + id‘klflfld) =0 01 = f’($0)-

i=1 j=l

 

Similarly,

f(fb'o) " f(xo - d)
d

 

 

oo . J;

= — 24-4-1 — Edi-13.9) =0 —9. = 76.).
i=1 j=l

Combining the above two equations, we obtain that

f(x +dl—f($0)_ 1 _ f(M)—f(x -d)
0 d —of($0)—0 d 0 -

 

 

Now assume that (f(xo + d) — f(xo))/d and (f(xo) — f(xo - d))/d are both at
most finite in absolute value, and their real parts agree. Then, using Equation (7.7),
mg; mar-1 + 21;, dqj‘lA,(d)l and | — 3:16.794 — 2,451, d‘i‘lBJ-(dfl are both at

most finite, and
co . J,- 00 . J1
Ems-1+ quf‘lAJ-(d) =0 - 26.-r1 — Zd‘i‘lBJ-(d).
i=1 j=1 i=1 j=1
Hence,
q, > 1 for allj E {1,...,Jr}, t,- > 1 for allj E {1,...,Jz}, and (11 = —61,

from which we infer, using Equation (7.8), that f is differentiable at x0 with

_ f($0+d)-f($0) _ f(xol—f($0-d)
‘0 d “0 d '

 

 

f’($0) = 011 = —,61

209

This finishes the proof of the first part of the theorem.

Since the second part of the theorem is only a special case of the last one, with
n = 2, we will go directly to proving the last part of the theorem. Note that since f

is (n — 1) times differentiable at x0,

f(xo+x) = Zf(t(.x0)x +2073 +Zx‘b‘A,

f(xo—x) = 53(— 1) +26.x+2xt’B

i=0 i=n
for 0 < x < a, where a is a positive real number, where the Aj’s and the Bj’s are as
before, and where the qj’s and the tj’s are noninteger rational numbers greater than

n—l.

Assume f is n times differentiable at x0. Then
q,- >nfor allj E {1,...,Jr}, t, >nfor allj E {1,...,Jz},

and
n! an = (-—1)"nl 6,, = f(")(xo).

It can be shown by induction on n that

d“"(zn:(—1)"‘j ( ’3? ) f(xo + 30)) =o n! a, and

i=0

71-" (i604 ( 3? ) f(xo —jd)] =0 <—1)"n! 6..

Therefore,

d-n (i(‘1)n_j ( T; ) f(330 +jdl) =0 f(n)($0)

j=0

:0 d‘“ (i(A)" ( 1; ) f(mo — J'dl) '

i=0

 

210

Now assume that

dr(§(—1)"-i(" )f(x0+jd)‘")andd (2H 1) (j)f (xo—jd))

are both at most finite in absolute value, and their real parts agree. Then
q,- > n for all j E {1,...,Jr}, t,- > n for all j E {1,...,Jz}, and n! (1,, = (-—1)"n! 6",

from which we infer, again using Equation (7.8), that f is n times differentiable at
1130 With

f(")(xo) = n! (1,, = (—1)"n! 6,, =0 d‘" (i(—1)""j ( 3' ) f(xo +jd))

:0 d‘" (g(-1V ( T; ) f(xo —jd)) -

This finishes the proof of the theorem.

Since knowledge of f (x0 — d) and f (x0 + (1) gives us all the information about a
computer function f extendable to x0 :1: d, in a real positive radius 0 around x0, we
have the following result which states that, from the mere knowledge of f (x0 — d)
and f (x0 + d), we can find at once the order of differentiability of f at x0 and the

accurate values of all existing derivatives.

Theorem 7 .5 Let f be a computer function that is continuous at x0 and extendable
to x0 :l:d. Then f is n times differentiable at x0 if and only if there exist real numbers
(11, . . . ,an such that

f(xc — d) =n f(370) + Z(-1)jajdj and f(xo + 61) =n f(mol + Zajdj-

Moreover, in this case f(j)(xo) = j! (1,- for 1 S j S n.

 

211

7 .4 Examples

As a first example, we consider a simple function and study its differentiability at
0. Let f (x) = x |x| + exp(x). It is easy to see that f is differentiable at 0 with
f (O) = f’ (0) = 1 and that f is not twice differentiable at 0. We will show now how
using the result of Theorem 7.4 will lead us to the same conclusion. First we note

that f is defined at 0 and extendable to id.

It is useful to look at what goes on inside the computer for this simple example.
Altogether, we need six memory locations to store the variable, the intermediate

values, and the function value. These six memory locations are

$1 5'1 = abstv), 5'2 = Sqrt(51),
S3 = x at 52, S4 = exp(x), a = S3 + S4.

So we can look at F( f) as a function from R6 into R6. Let

[EzR—+ R6; E(x) = (x,0,0,0,0,0)

] F I R6 —’ R6; F(31p21p31P41P51P6) = ($1811521331341a)
PIRG—tR; P($,Sl,Sz,S3,S4,a)=a .

[GzR—->R; G(x)=PoFoE(x)

 

Then G (x) = a =M f (x), where M is an upper bound of the support points that can

be obtained on the computer.

If we input the value x = —d, then the six memory locations will be filled as

follows:

1‘ = -d, 51 = d, S2 = (ll/2,

53 = -d3/2, S4 = fiu—lrdj/fl. a = -33/2 + z£o(—1)jdj/j!.

So the output will be G(—d) = 1 -d—d3/2+d2/2!+Ejhi3(—1)jdj/jl =M f(—d). If we
input the value x = 0, the output is G(O) = 1. Since f (0) is real and f (0) =M G(O),

 

212

we infer that f (0) = 1. Similarly, we find that

G(d) = 1+d+d3/2+d2/2!+§:dj/j!=M f(d)
j=3

G(—2d) = 1— 2d — 23/2413/2 + 242 + §(—2)jdj/j! =M f(-2d)

i=3

M
G(2d) = 1+ 2d + 23/2d3/2 + 2012 + Z 2361' /j! =M f(2d).

j=3
Note that f (—d) =0 1 = f (0) =0 f ((1); hence f is continuous at 0. Since

1(4) — 7(0) :0 1 :0 7(0) — f(-d)
d d ’

 

 

we infer that f is differentiable at 0, with f’ (O) = 1. However,

f(2d) — 25(4) + 7(0) ___0 (23,2 _ 2) ,4). + 1,

which implies that I( f (2d) - 2 f (d) + f (0)) / d2] is infinitely large. Hence f is not twice

differentiable at 0.

Next, we consider the two functions already mentioned in the introduction, Equa-
tion (7.2), which are clearly computer functions. Consider first the function 91 (x) If
we input the values x = -3d, —2d, —d, 0, d, 2d, 3d, we obtain the following output up

to depth 3

91(i3d) =3 15.66398831641272d5/2
91(i2d) =3 5.684263512907927d5/2

gl(id) =3 1.004845319007115d~'>/2

91(0) = 0'

Since g1(—d) =0 g1(0) =0 gl(d), g1 is continuous at 0. A simple computation shows

that {g1(d) — 91(0)}/d =0 0 =0 {g1(0) — gl(—d)}/d, from which we infer that 91 is

 

213

differentiable at 0, with g’,(0) = 0. Also {g1(2d) — 2g1 (d) + 91 (0)} /d2 =0 0 =0 {g1(0) —
Zgl(-—d) + g1(—2d)}/d2, from which we conclude that g1 is twice differentiable at 0,
with g[2)(0) = 0. On the other hand, {g1 (3d) — 3g1(2d) + 3g1(d) — 91 (0)} /d3 ~ d‘l/z,
which entails that |(gl(3d) — 3g1(2d) + 391((1) — 91 (0)) /d3| is infinitely large. Hence

g1 is not three times differentiable at 0.

By evaluating g2(-d) and g2(d) up to any fixed depth and applying Theorem 7.5,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Order n gén)(0) CPU Time
0 0. 3.400 msec
1 1.004845319007115 4.030 msec
2 0.9202876179268508 5.710 msec
3 —18.81282866172102 8.240 msec
4 —216.8082597872205 12.010 msec
5 —364.2615904917884 17.570 msec
6 101933.1724529188 25.150 msec
7 3798311370563978 35.700 msec
8 6076535384260825 49.790 msec
9 —1441371402.871872 67.210 msec
10 -156736847166.3961 89.840 msec
11 —6725706835826.155 118.950 msec
12 —131199307184575.8 154.530 msec
13 5770286440090848. 200.660 msec
14 0.7837443136320079 x 1018 256.460 msec
15 0.4850429351252696 x 1020 321.630 msec
16 0.1734774579876559 x 1022 400.140 msec
17 —0.1757849296527536 x 1023 478.940 msec
18 —0.9350429649226352 x 101’5 582.150 msec
19 —0.9521402181303937 x 1027 702.390 msec

 

 

 

Table 7.4: g§")(0), O S n S 19, computed with R calculus

we obtain that g2 is differentiable at 0 up to arbitrarily high orders. In Table 7.4,
we list only the function value and the first nineteen derivatives of g2 at 0, together
with the CPU time needed to compute all derivatives up to the respective order.
The numbers in Table 7.4 were obtained using the implementation of R in COSY

INFINITY [8, 12].

 

214

7.5 Computer Functions of Many Variables

Since we know now how to compute the nth order derivative of a real computer
function of one variable at a given real point x0 whenever the nth order derivative
exists and the function is extendable to x0 :t d, the following lemma shows how to find
all nth order partial derivatives at a given real point p}, of a function f : Rm —» R which
can be represented on a computer whenever all the nth order partial derivativex exist

and are continuous in a neighborhood of 50 and extendable to fig + (:lzd, :lzd, . . . , id).

Lemma 7 .4 Let f : R'" —> R be a function representable on a computer whose
nth order partial derivatives exist and are continuous in the neighborhood of the point
p}, = (x01,x02, . . . , xOm). Then the nth order partial derivatives of f at p}, can always be

computed in terms of nth order derivatives of real computer functions of one variable.

Proof. Let I be the number of nth order partial derivatives of f; we note in passing
that it can be shown [2] by induction on n and m that l = (n+m-1)!/(n! (m -— 1)!).
Let k = l-m and let p1,p2,...,p1c denote the first k prime numbers. For j = 1, . . . , k,

let a,- = n+\1/p_,-. For i = 1,...,l, let
film) = f(2701 + (1(.-_.1)m+1x, $02 + a(i—l)m+2$1-°-1xOrn + 011mm).

Then f,-,i = 1, . . . , l, are 1 real computer functions of x, n times differentiable at

0. Evaluating ((1” f,- /dx")[$__,0 for i = 1, . . . ,l yields l equations in the l unkowns

 

aflf th n113’21H-1nme {O,1,...,n}
017'“ (3le2 anm 1 1’ W1 an
11 21 1 m P=P0 n1+n2+...+nm:n

The matrix M of the coefficients has as entries products of the different (1’s raised

to exponents between 0 and n. In the ith row, we have only products of the form

 

215

Cn;n1,n2....,nma(il_1)m “(12:11,", +2 . “(13:2, where Cn;n1.n2.....nm is a positive integer. The
determinant of M is the sum of l! terms, each of which is the product of a positive
integer and the (1’s raised to exponents less than or equal to n, and such that not
all the exponents in any one term agree with those in any of the remaining (l! — 1)
terms. By our choice of the a’s, no cancellation in the evaluation of the determinant

could occur; hence det M 7E 0.

It is worth noting that the choice of the (1’s above is far from being the only one
possible. Let a1,(12, . . . ,ak be any set of k real numbers. We look at detM as a
function from R" into R. A purely statistical argument shows that it is very unlikely
that det M be zero for a given choice of numbers. We are led to believe that there
exist even uncountably many choices of ((11, a2, . . . , (11,) E R" that give a nonvanishing
determinant. Here we provide simple choices of the (1’s only in the case m = 2: For
m = 2, we have that l = n+1 and k = 2(n+1). Fori = 1,2,...,n+ 1, let (12,-_1 =1
and (12,- = 6.1, where 60 = O and 6,, 75 6,, if jl 7t jz in {0,1,...,n}. Then

1 0 0 O 0
M: 1 n61 £3,226? n61"1 fl?
1 n6, Jug—163; 7163-1 6;;
Therefore,
61 6: fl?
detM=Cn 6? 3:2 '33 =CnV(61,62,...,6,,), where
3.. 53. 3.?

 

 

011:1:1 ( T; )1 and V(18111621--~1,Bn)= (f1 31:1) 1:1 f1 (fika —,Bk2)

J 1 k1=1 k2=1 k3=lc2+1

 

216

is the well-known Vandermonde determinant. Hence

deth (11 (3.7: )) (f1 [3,-2) 11 fi (fin—3.0750.

ji=1 j2=1 j3=1 j4=ja+1

Bibliography

[1]

[2]

[3]

[4]

[5]

[6]

N. L. Alling. Foundations of Analysis over Surreal Number Fields. North Holland,

1987.

M. Berz. Diflerential algebraic description of beam dynamics to very high orders.
Particle Accelerators, 24:109, 1989.

M. Berz. Automatic differentiation as nonarchimedean analysis. In Computer
Arithmetic and Enclosure Methods, page 439, Amsterdam, 1992. Elsevier Science
Publishers.

M. Berz. COSY INFINITY Version 6 reference manual. Technical Report
MSUCL—869, National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, 1993.

M. Berz. Analysis on a N onarchimedean Extension of the Real Numbers. Lecture
Notes, 1992 and 1995 Mathematics Summer Graduate Schools of the German
National Merit Foundation. MSUCL—933, Department of Physics, Michigan State

University, 1994.

M. Berz. COSY INFINITY Version 7 reference manual. Technical Report
MSUCL—977, National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, 1995.

217

 

218

[7] M. Berz. Calculus and Numerics on Levi-Civita Fields. In M. Berz, C. Bischof,
G. Corliss, and A. Griewank, editors, Computational Differentiation: Techniques,

Applications, and Tools, pages 19—35, Philadelphia, 1996. SIAM.

[8] M. Berz. COSY INFINITY Version 8 reference manual. Technical Report
MSUCL-1088, National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, 1997.

[9] M. Berz. Modern Map Methods in Particle Beam Physics. Academic Press, San
Diego, 1999.

[10] M. Berz. Nonarchimedean Analysis and Rigorous Computation. International

Journal of Applied Mathematics, submitted and accepted for publication in 2000.

[11] M. Berz, C. Bischof, A. Griewank, and G. Corliss, Eds. Computational Differ-
entiation: Techniques, Applications, and Tools. SIAM, Philadelphia, 1996.

[12] M. Berz, G. Hoffstiitter, W. Wan, K. Shamseddine, and K. Makino. COSY
INFINITY and its Applications to Nonlinear Dynamics. In M. Berz, C. Bischof,
G. Corliss, and A. Griewank, editors, Computational Differentiation: Techniques,

Applications, and Tools, pages 363-365, Philadelphia, 1996. SIAM.
[13] J. H. Conway. 0n Numbers and Games. North Holland, 1976.
[14] M. Davis. Applied Nonstandard Analysis. John Wiley and Sons, 1977.
[15] H.-D. Ebbinghaus et a1. Zahlen. Springer, 1992.

[16] L. Fuchs. Partially Ordered Algebraic Systems. Pergamon Press, Addison Wesley,
1963.

[17] H. Gonshor. An Introduction to the Theory of Surreal Numbers. Cambrindge

University Press, 1986.

 

 

219

[18] H. Hahn. Uber die nichtarchimedischen Gréfiensysteme. Sitzungsbericht der
Wiener Akademie der Wissenschaften Abt. 2a, 117:601—655, 1907.

[19] E. Hewitt and K. Stromberg. Real and Abstract Analysis. Springer, 1969.

[20] D. E. Knuth. Surreal Numbers: How two ex—students turned on to pure mathe-

matics and found total happiness. Addison-Wesley, 1974.

[21] D. Laugwitz. Eine Einfiihrung der Delta-Funktionen. Sitzungsberichte der Bay-
erischen Akademie der Wissenschaften, 4:41, 1959.

[22] D. Laugwitz. Anwendungen unendlich kleiner Zahlen 1. Journal fiir die reine
und angewandte Mathematik, 207:53—60, 1961.

[23] D. Laugwitz. Anwendungen unendlich kleiner Zahlen 11. Journal fiir die reine
und angewandte Mathematik, 208:22—34, 1961.

[24] D. Laugwitz. Eine nichtarchimedische Erweiterung angeordneter K6rper. Math-
ematische Nachrichten, 37:225—236, 1968.

[25] D. Laugwitz. Ein Weg zur Nonstandard-Analysis. Jahresberichte der Deutschen
Mathematischen Vereinigung, 75:66—93, 1973.

[26] D. Laugwitz. Tullio Levi-Civita’s work on nonarchimedean structures (with an
Appendix: Properties of Levi-Civita fields). In Atti Dei Convegni Lincei 8:
Convegno Internazionale Celebrativo Del Centenario Della Nascita De Tullio

Levi-Civita, Academia Nazionale dei Lincei, Roma, 1975.

[27] Tullio Levi-Civita. Sugli inflniti ed infinitesimi attuali quali elementi analitici.

Atti Ist. Veneto di Sc., Lett. ed Art., 7a, 4:1765, 1892.

[28] Tullio Levi-Civita. Sui numeri transfiniti. Rend. Acc. Lincei, 5a, 7:91,113, 1898.

 

 

220

[29] A. H. Lightstone and A. Robinson. Nonarchimedean Fields and Asymptotic
Expansions. North Holland, New York, 1975.

[30] L. Neder. Modell einer Leibnizschen Differentialrechnung mit aktual unendlich
kleinen Grfiflen. Mathematische Annalen, 118:718—732, 1941-1943.

[31] William Fogg Osgood. Functions of Real Variables. G. E. Stechert & CO., New
York, 1938.

[32] A. Ostrowski. Untersuchungen zur arithmetischen Theorie der K6rper. Mathe-
matische Zeitschrift, 39:269—404, 1935.

[33] LB. Ball. The arithmetic of Differentiation. Mathematics Magazine, 59:275,
1986.

[34] A. Robinson. Non-standard analysis. In Proceedings Royal Academy Amsterdam,
Series A, volume 64, page 432, 1961.

[35] A. Robinson. Non-Standard Analysis. North-Holland, 1974.
[36] W. Rudin. Real and Complex Analysis. McGraw Hill, 1987.

[37] C. Schmieden and D. Laugwitz. Eine Erweiterung der Infinitesimalrechnung.
Mathematische Zeitschrift, 69:1—39, 1958.

[38] K. Shamseddine and M. Berz. Exception Handling in Derivative Computa-
tion with Non-Archimedean Calculus. In M. Berz, C. Bischof, G. Corliss, and
A. Griewank, editors, Computational Differentiation: Techniques, Applications,
and Tools, pages 37—51, Philadelphia, 1996. SIAM.

[39] K. Shamseddine and M. Berz. Non-Archimedean Structures as Differentiation
Tools. In Proceedings of the Second LAAS International Conference on Computer
Simulations, pages 471—480, Beirut, Lebanon, 1997.

 

221

[40] K. Shamseddine and M. Berz. The N on-Archimedean Field R, Overview and
Applications. In Proceedings of the International Conference on Scientific Com-

putations, Beirut, Lebanon, 1999.

[41] K. Shamseddine and M. Berz. Power Series on the Non-Archimedean Field R. In
Proceedings of the International Conference on Scientific Computations, Beirut,

Lebanon, 1999.

[42] K. Shamseddine and M. Berz. The Differential Algebraic Structure of the Levi-

Civita Field. Journal of Symbolic Computation, submitted.

[43] K. Shamseddine and M. Berz. Power Series on the Levi-Civita Field. Interna-
tional Journal of Applied Mathematics, submitted and accepted for publication
in 2000.

[44] K. Stromberg. An Introduction to Classical Real Analysis. Wadsworth, 1981.

[45] K. D. Stroyan and W. A. J. Luxemburg. Introduction to the Theory of Infinites-
imals. Academic Press, 1976.

Index

=,., 35
Q, 1, 18
62+, 18
Q‘, 18
R, 1, 18
R+, 18
R‘, 18
Z, 18
Z+, 18
Z‘, 18
z, 35
cos, 89
cosh, 90
cot, 95
exp, 89
>>, 47
A, 35
<<, 47
(9, 43
~, 35
sin, 89
sinh, 90

222

tan, 94

d, 36

n—times differentiability, 128
rules about, 131

nth derivate function, 128

nth derivative, 128

 

of computer functions, 195
R, 32

a field, 38

a non—Archimedean field extension
of R, 32

addition and multiplication on, 37

algebraic structure of, 32

calculus on; see Calculus on R, 6

Cauchy complete in order topology,
60

definition of, 34

different from R, 63, 65, 99, 112—
114,128

differential algebraic structure of, 43

embedding of R in, 37

existence of roots in, 41

223

expandable functions on, 167

is the Levi-Civita field, 4

not Cauchy complete in weak topol-
ogy, 74

order structure of, 45

power series on, 82, 167

strong convergence in, 58

t0pological structure of, 49

uniqueness of, 55

Absolute value, 49
Algebraic numbers, 1
Alling, N. L., 4
Analysis, 1
Non-Archimedean; see Non-Archimedean
Analysis, 4
Nonstandard; see Nonstandard Anal-
ysis, 3
Automorphism
field; see Field automorphism, 23

group; see Group automorphism, 27

Berz, M., 4

Calculus
non-Archimedean; see Non-Archimedean
calculus, 98
Calculus on R, 6, 98, 196

Calculus on the expandable functions,
174
Cauchy, 40
strongly; see Strongly Cauchy, 60
weakly; see Weakly Cauchy, 74
Cauchy complete, 1, 40, 60, 74
Cauchy sequence, 40
Chain rule, 110, 123
Composition

of continuous functions, 118

 

of differentiable functions, 123 F
of expandable functions, 170
of topologically continuous functions,
102
of topologically differentiable func-
tions, 110
Computational differentiation, 2, 6, 197
Computer functions, 6, 194
n-times differentiable, 207, 210
computation of derivatives of, 7, 195,
205
example, 211, 213
continuation from R to R, 204
continuous, 206
differentiable, 207
extendable, 199, 206

local form of, 200
machine level representation of, 198
of many variablas, 214
of one variable, 197
Continuation
of computer functions, 204
of real functions, 4
of real power series, 89, 199
Continuity, 115
=>boundedness, 115
=>topological uniform continuity, 115
rules about, 117
topological; see Topological conti-
nuity, 99
Continuum, 2
Contracting function, 40—42, 162, 180,
183
Convergence
strong; see Strong convergence, 57
weak; see Weak convergence, 53
Conway, J ., 4, 8
COSY INFINITY, 195, 213
Criterion for strong convergence
of infinite series, 64
of power series, 83

of sequences, 61

224

Criterion for weak convergence
of power series, 86

of sequences, 72

Davis, M., 3

Delta functions, 3

Derivate function, 107, 120
nth, 128 f

Derivation, 43

Derivative, 107, 120
nth, 128

 

computation of, 205
of a composition, 110, 123
of a product, 109, 122
of a sum, 109, 122
of computer functions, 195
Derivatives are differential quotients, 2,
6, 121
Diflerentiabifity, 120
=> topological differentiability, 120
n-times, 128
infinitely often, 132
rules about, 122
test for, 125
topological; see Topological differ-
entiability, 107
Differential, 47

225

Difl'erential Algebras, 13
Differential quotients, 2, 48, 121

Distributions, 2

Examples
R not Cauchy complete in weak t0pol-
ogy, 74
computation of derivatives of com-
puter functions, 211—213
function on a closed interval
topologically continuous; multiple
primitives, 114
topologically continuous; no max-
imum, 1 14
topologically continuous; not all
intermediate values assumed, 113
topologically continuous; not bounded,
99
topologically differentiable; mul-
tiple primitives, 114
topologically differentiable; no max-
imum, 114
infinitely often differentiable, 134
infinitely often differentiable; Tay-
lor series converges nowhere to

the function, 134

topologically differentiable with zero
derivative; not constant, 113
topologically diflerentiable; not all
intermediate values assumed, 113
nontrivial field automorphism on R,
47
nontrivial order preserving field au- [
tomorphism on L, 31

strong convergence of power series

undecided, 85

 

Expandable functions, 167 L
algebraic pr0perties of, 168
calculus on, 174
definition of, 167
infinitely often differentiability of,

185
integration of, 190
intermediate value theorem for, 175,
184, 185
maximum theorem for, 186, 187
mean value theorem for, 189

Rolle’s theorem for, 188

Field automorphism, 23, 24, 56
only one on Q, 25
only one on R, 26, 47, 56

order preserving, 25

on L, 28
on R, 28
Finite, 47
Fixed point theorem, 38, 41, 42, 160,
183
Formula manipulators, 194, 196
Frobenius, 46
Function
nth derivate, 128
computer; see Computer functions,
194
contracting; see Contracting func-
tion, 40

delta; see Delta functions, 3

derivate; see Derivate function, 107

expandable; see Expandable func-
tions, 167

Heaviside, 197

intrinsic, 197

quasi—linear; see Quasi-linear func-
tion, 147

reciprocal, 199

root, 199

transcendental; see Transcendental

functions, 1

Game theory, 4

226

Geometry, 1
Gonshor, H., 4

Group automorphism, 27

Hahn theorem, 8

Heaviside function, 197

Implementation, 13, 35, 53, 206
Index of a function, 173
Infinitely large, 2, 47
Infinitely often differentiable, 133, 185
Infinitely small, 2, 47
Infinitesimal, 36, 47
Integration, 190
Intermediate value theorem, 6, 7, 113,
147, 157, 175, 184—186
Intrinsic functions, 197
Inverse function, 23, 164
Inverse flmction theorem, 147, 164
Isomorphism, 22—24
order preserving, 22, 24

Iteration, 40, 157

Knuth, D., 4

Laugwitz, D., 3
Laws of nature, 1

Left-finite sets, 33

properties of, 33

 

Leibniz, 48
Levi-Civita, 4, 32
Lightstone, A. H., 4
Lipschitz constant, 115

Luxemburg, W. A. J., 3

Mathematica, 195, 196
Maximum theorem, 6, 7, 186, 187
Mean value theorem, 7, 189
Measure topology, 54

Measurement, 1

Non-Archimedean Analysis, 4, 5
Non-Archimedean calculus, 7, 98, 196
Non-Archimedean field, 2, 7, 21, 22, 25,
27,32,34,47,54,56,128,194,
197
Nonstandard Analysis, 3, 13
Numbers
algebraic, 1
rational, 1
real, 1
surreal, 4, 8

Numerical differentiation, 195

Order topology, 5, 5O
Ostrowski, A., 4

227

Power series, 1, 5, 82, 140, 143, 167,
198

Continuation from R to R, 198
radius of convergence of, 86, 144
reexpansion of, 144
strong convergence of, 83
weak convergence of, 86, 167
with purely real coefficients, 89, 198

radius of convergence of, 89, 198

Quasi-linear function, 147
intermediate value theorem for, 157
inverse function theorem for, 164
properties of, 151, 153, 154
scaled function of, 151

properties, 151, 153, 154
strictly increasing, 155

strictly monotone, 155

Radius of convergence of power series,
86, 144, 167
with purely real coefficients, 89, 198
Rational numbers, 1
Real numbers, 1
Reexpansion of power series, 144
Regular sequence, 57, 62, 83, 167

Remainder formula

 

of order 0, 116
of order 1, 121
of order n, 130
Robinson, A., 3
Rolle’s theorem, 7, 188
Roots, 1, 3, 5, 22, 34, 36, 41, 199

Schmieden, C., 3
Secants, 2, 128, 129, 147
Semi-norm, 51, 70
Sequence
Cauchy; see Cauchy sequence, 40
regular; see Regular sequence, 57
strongly Cauchy, 60
strongly convergent, 58
weakly Cauchy, 74
weakly convergent, 70
with purely real members, 69, 77

Skeleton group, 19

S], = Z, 20
SR ={0}120
SR = Q1 21

group automorphism on, 27
Strong convergence, 57

=> weak convergence, 81

of infinite series, 64, 65

criterion, 64

228

of power series, 83
criterion, 83
of sequences, 58
criterion, 61
Strongly Cauchy, 60
Strongly convergent, 58, 62
=> regular, 62
Stroyan, K. D., 3
supp, 35
Support, 35

Surreal numbers, 4, 8

The number d, 36
Theorem
fixed point, 38, 41, 42, 160, 183
Hobenius, 46
Hahn, 8
intermediate value, 6, 7, 113, 147,
157, 175, 184—186
inverse function, 147, 164
maximum, 6, 7, 186, 187
mean value, 7, 189
Rolle’s, 7, 188
Topological continuity, 6, 99
at a point, 99
on a set, 99

rules about, 101

T0pological differentiability, 6, 107, 204
=>topological continuity, 108
at a point, 107, 204
on a set, 107
rules about, 109
Topological uniform continuity, 103
Topology
induced from R into R, 50, 53, 54,
68, 77
measure, 54
order, 50
weak, 53

Transcendental functions, 89

Uniqueness of R, 1

Uniqueness of R, 55

Weak convergence, 53, 69, 72, 167
of infinite series, 78
of power series, 86, 167
criterion, 86
with purely real coefficients, 89
of sequences, 70, 72
criterion, 72
Weak topology, 5, 53
Weakly bounded, 75
Weakly Cauchy, 74

229

Weakly convergent, 70

”llllllllllllllf