a: nut]
‘.. . hfiui-

Lufiwuwnmmm..m
- H I O

.-
155. v o

‘Qflp t. ft 3.1..

.f
..A. 1|.vov1
J .1.

1.0.3 I’obfi.
ul‘IIJ.‘ i'vf.
FOIV’?‘ .I .c»
1.1.5.7..
Y;

7 .

.5:-
va'AaIIIco

1"},

' 31.9..l.
ex . t

h

 

L
Brit's I: 9..
. . .. :c} .5.
‘V 17‘3.’: I... I V 0' '1’.
Ir... agatzhi... ;
u .r. .L. 7125:: 2’
.i

. at
y. .39.:

 

2,1311!!! . 1
XI: :13: I III“!
.i. .51.... s If ‘11 .ll

Ii..é;c.l .rla nun-“h..4t oiled-IO: I... I. v I! ’9!!!”

i.‘1t.§ar‘l.£ .n.£....:.ftlra§.-.!
LIL .v«..ll.... . I.
x .
. 1 .
v-slvh‘ll", 71’

 

.J In!"

'HFSlfl

lllllllll\llllllllllllHlllllllll llllllljlll

31293 0079 91

 

LIBRARY
‘ Mich'Qan State
nlverslty

 

 

This is to certify that the

dissertation entitled

THE t’EPxNO DEK‘WA'UVES

presented by
HMfwm N F E32; \ C

has been accepted towards fulfillment
of the requirements for

Pill. D degreein ma‘f’tlepMCti‘l‘Cs

 

 

owed a my

”1/ Major professor

 

DateOdsywa

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

THE PEANO DERIVATIVES
by

Hajrudin Fejzié

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1992

Let }
has a it!
that fur
The (on
l‘th 0rd
but the

Lot
by .X' a
have sh
ditteron
deriva:
sets .4"
they p
the firs
that {(
shown
Peano

A.
ll\'eg_
Ofdln.‘
tires

Wit

ABSTRACT
THE PEANO DERIVATIVES
by
Hajrudin Fejzié

Let f be a function defined on an interval [a,b] and that k 6 N. We say that f
has a Ic-th Peano derivative at :1: E [a, b] if there exist coefficents f1(x), . . . , fk(:c) such
that f(a:r+ h) = f(x) + hf1(:r) + . . . + %fk(a:) + hkek($, h) where limh_.o ek(:c, h) = O.
The coeflicent fk(:r) is called the k-th Peano derivative of f at x. The existence of a
k-th ordinary derivative, f(")(:r), implies the existence of fk(:c) and fk(:c) = fl")(:z:),
but the converse is not true for k 2 2.

Let A’ be the class of all derivatives, and let [A’] be the vector space generated
by A’ and O’Malley’s class Bf. S. Agronsky, R. Biskner, A. Bruckner and J. Marik
have showed that every function [A’] has the form g’ + hk’, where g,h and k are
differentiable. They also proved that f 6 [A’] if and only if there is a sequence of
derivatives {22“} and closed sets {An} such that U§°=1An = R and f = vn on A... The
sets An and corresponding functions 1),. are called a decompositon of f. The question
they posed is whether every Peano derivative belongs to this class of functions. In
the first part of this thesis a. positive answer to this question is given. Also it is shown
that for Peano derivatives the sets An can be chosen to be perfect. Moreover it is
shown that every k—th Peano derivative is the composite derivative of the (k — 1)-th
Peano derivative relative to the sequence {An}.

A. Bruckner, R. O’Malley and B. Thomson introduced the notation of path deriva-
tives. They showed that path derivatives have many of the properties possessed by
ordinary derivatives. In the second part of this thesis it is shown that Peano deriva-
tives are also path derivatives and hence they have all the properties possessed by

path derivatives. This gives another proof of the many properties possessed by Peano

'
.n "‘ 'l .‘l’lfl’fi

 

 

 

 

derive
Tl
derivz
quest
Fl
Peanc

and s

derivatives and also answers the question posed by the above authors.

The third part of this thesis shows that a k-th Peano derivative is a selective
derivative of the (k - 1)-th Peano derivative, and hence gives a positive answer to the
question posed by C. Wei] regarding Peano and selective derivatives.

Finally the last part of this thesis shows that these results are still true if we replace
Peano derivatives with generalized Peano derivatives, introduced by M. Laczkovich,

and studied by C. Lee.

To my brother-in-law MIDHAT DRINA, who was killed by Serbian irregulars on
June 22, 1992.

iv

ACKNOWLEDGMENTS

I would like to thank my advisor, Professor Clifford Weil, for all his help,

encouragement and advice. His knowledge and enthusiasm were invaluable.

TABLE OF CONTENTS

INTRODUCTION .......................................................... 1
CHAPTER I .............................................................. 10
1.1 Decomposition of Peano derivatives ................................. 10
CHAPTER II ............................................................. 30
2.1 Peano and path derivatives ......................................... 30
CHAPTER III ............................................................ 41
3.1 Relationship between fk and f)._1 ................................... 41
3.2 Peano derivatives and Property Z ................................... 48
3.3 Peano and selective derivatives ...................................... 51
CHAPTER IV ............................................................ 57
4.1 Decomposition of Generalized Peano derivatives ..................... 57
4.2 Generalized Peano, path and selective derivatives .................... 68
BIBLIOGRAPHY ......................................................... 76

vi

 

 

 

 

The

in he
defi\'
to fir
prevh
One

Inofix

Pean<

Defir

crfit

uhrrg

f 011

17
introc
Poirm
Innst;

fin k

u
¢

It iSe
defiVaI

thf‘re i

INTRODUCTION

The definition of the k-th ordinary derivative of a real valued function is iterative
in nature and thus easily comprehended if one initially understands what a first
derivative is. This nice feature can present a problem, however, because in order
to find the k-th derivative of a function f at a point 1:, one must know all the
previous derivatives, not only at x, but at every point in some neighborhood of 3:.
One type of generalized k-th order differentiation, having Taylor’s theorem as its
motivation, attempts to skirt this drawback. This kind of differentiation is called

Peano differentiation.

Definition 0.0.1 A function f is said to have a lc-th Peano derivative at x if there
exist numbers f1(a:), f2(a:), . . . ,fk(:c) such that

k
f(1 + h) = ftx) + W) + - - - + $4142) + ektx. h» (1)

where 6;.(3, h) —+ 0 as h —) 0. The number fk(:c) is called the k-th Peano derivative of
f at at. It will be convenient to denote f (2:) by fo(x). With this notation (1) becomes

k hi h"
f(1' + h) = Z fifflx) + Felix, h)-
j=o ° '
This concept was presented in 1891 by the italian mathematician G. Peano. Peano
introduced this type of derivative, obtained a product rule, a quotient rule, and
pointed out that if a function f has an ordinary k-th derivative at :c, f(*)(z), then it

must have a k-th Peano derivative at a: and fk(:r) = f‘k)(z). The converse is not true

for k 2 2 as can be seen from the following example. Let
f(x) = 16"“H sin-i: for a: 76 0 and f(0) = 0.

It is easy to see that fk(0) = 0 but f(") at 0 doesn’t exist. Thus the k-th Peano
derivative is a true generalization of the ordinary k-th derivative although obviously

there is no difference for k = 1.

~.:'-

 

 

 

 

In
exhib
derive
tive.
Barre
funrti

where
the D.
and f,

and e;

Cliftcr
flinch}

fl: p0,:

dem'at

Defini
intf'f‘t‘q

FE Wing

In 1954 H. W. Oliver published the first extensive work devoted exclusively to
exhibiting properties of k-th Peano derivatives. (See [10].) He showed that such a
derivative has several of the pr0perties known to be possessed by an ordinary deriva-
tive. Oliver established that if f], exists for all :1: in some interval I, then fk is of
Baire class one; i.e., f). can be written as a pointwise limit of a sequence of continuous
functions (A). (Denjoy had obtained this result earlier in the more general setting
where fl. is defined relative to a perfect set H.) Oliver also showed that f]. must have
the Darbouz property; i.e., that for any interval [a, b] C I if y is a point between fk(a)
and fk(b), then there is c 6 (a, b) so that fk(c) = y (B), another property well known
and easily verified for ordinary derivatives. Moreover, he showed that if f]. is bounded
above or below on some [a, b] C I, then f]. is the ordinary lc-th derivative off on [a, b]
(C). In particular, this yields the monotonicity theorem which states that if fk Z 0 on
[a,b], then fk_1 is nondecreasing and continuous on [a, b] (D). Combining this with
the fact that fk is of Baire class one, it follows that fk is an ordinary k-th derivative
on an open, dense subset of I (E). R. J. O’Malley and C. E. Weil showed that if
fk attains both values —M and M on some interval [a, b] C I, then there is an open
interval J C [a,b], on which f1, = f“) and fl") attains both values —M and M on J
(F). (See [12].)

Ifg is an ordinary derivative on I, then for any open interval, (a,b), g‘1(a,b)
either is empty or has positive Lebesgue measure, a result first proved by Denjoy. A
function having this property is said to have the Denjoy property. Oliver showed that
fk possesses the Denjoy property on I (G). (See [10].)

Z. Zahorski proved that the following property is possessed by every ordinary

derivative.

Definition 0.0.2 A function g is said to have the Zahorski property if for each open
interval (a,b), for each a: E g‘1(a,b), and for each sequence of intervals {In} con-

verging to 2:, (The end points of the In converge to a: but 2: belongs to no In.) with

 

 

 

 

Defin
for er:

that g'

“I
still is

In
Proacl
genera
merit i<
look-in

Th

alapt

for ap:
anolho
dm...,
{Unclio

fOraD

m(g"1(a, b) n I") = 0 for every n, implies limn_.o‘> 3% = 0, where m(I,,) denotes

Lebesgue measure of In and dist(:r, 1,.) denotes the distance between a: and In (H).

C. E. Weil showed that a k-th Peano derivative also has the Zahorski property,
and he introduced a property somewhat stronger that the Zahorski property, which
he called property Z. (See [18].)

Definition 0.0.3 A function g defined on an interval I is said to have property Z if
for every 6 > 0, each :2: E I, and each sequence of intervals {In} converging to a: such

that g(y) 2 g(m) on In or g(y) S g(x) on In for each n, we have

1. m({y e I. = lg(y) -9($)| 2 €}) = 0
n...» m(1n) + dist(a:,1,.) '

Weil showed that this property is strictly stronger than the Zahorski property, yet
still is possessed by every k-th Peano derivative.

In [3] the authors introduced the concept of a path derivative as a unifying ap-
proach to the study of a number of generalized derivatives. Namely since many other
generalized derivatives like approximate derivative, possess most of the properties
mentioned above that are possessed by Peano derivatives, the authors in [3] where
looking for a framework within which all of these derivatives could be presented.

The perspective they chose was to consider just those derivatives of a function F

at a point x which can be obtained as

 

Hm F (v) - F (x)

146E... 14-”: y — a:

for appropriate choices of sets Ex. One generalized derivative, then, differs from
another only by the choice of the family of sets {Ex : a: E R} through which the
difference quotient passes to its limit. For example, an approximately differentiable
function F permits a choice of sets {E3 : a: E R} so that each E: has density 1 at 2:;

for a Dini derivative each set may consist only of a sequence converging to m. This

framework includes any generalized derivative for which the derivative at a point is a
derived number of the function at that point. Since Weil has proved that fk(:v) is a
derived number of fk_1 at a point x, we see that this concept of path derivatives also
includes k-th Peano derivatives.

But in order to get some properties for path derivatives, like those possessed by
Peano or approximate derivatives, we require that the family of sets {E3 : :r E R}
satisfy various “thickness” conditions. These conditions relate to the “thickness” of
each of the sets E, and the way in which two of the sets intersect. The authors
proved that path derivatives with certain type of conditions imposed on the family of
sets {Ex : a: 6 R}, have many of the properties possessed by approximate and Peano
derivatives.

We will show that Peano derivatives are path derivatives with {E3 : a: E R} satis-
fying some of the intersection conditions introduced by the authors mentioned above.
This will give a positive answer to the question posed in [3]. In proving this assertion,
we won’t use any known results for Peano derivatives. So this can be regarded as
a new approach to studying Peano derivatives. Namely all of the properties (B),
(C), (D), (E), (F), (G) and (H), that we mentioned before, we will get for Peano
derivatives directly from the corresponding properties of path derivatives. The main

tool will be a decomposition of Peano derivatives which we will discuss next.

Let C be the family of all continuous functions on R, A the family of all differ-
entiable functions on R and A’ the family of all derivatives on R. If I‘ is a family of
functions defined on R, then by [I‘] we denote the family of all functions f on R with
the following property: for each n E N there exist vn E I‘ and a closed set A, such
that f = v" on An and ugg,A,. = R. In [1] (Theorem 2) it is shown that the following

four conditions are equivalent:
(i) There are g, h and Is: in A such that h’, k’ E [C] and f = g' + hk’.
(ii) There is a (p E A’ and d) E [C] such that f = (p + 1!).
(iii) The function f E [A’].

(iv) There is a dense open set T such that f is a derivative on T and f is a derivative

on R \ T with respect to R \ T.

The statement (ii) implies that [A’] is the vector space generated by A’ and [C].
In [1] (Theorem 3) it is shown that each approximate derivative, each approximately
continuous function and each function in Bf = [C] belongs to the class [A’]. In [10]
O’Malley showed that for approximate derivatives, the sets An from the definition of
[A’] can be chosen to be perfect. The following question is raised in [1]. “Does every
Peano derivative belong to [A’]?”. We will give a positive answer to this question,

plus we will prove even more.

Definition 0.0.4 Let f be a function defined on R. If there exist a function g, and
closed sets A", n = 1,2,. .. such that Ufi°=1An = R and gl’ "(3) = f(a:) for :c E An,

then we say that f is a composite derivative of g.

We will prove that f;‘ is a composite derivative of fk_1 with respect to the sets

P1,“, where for e > 0, 6 > 0 we define

P5 = P(f,e,6) = {2: : Iek(:c,h)| < e for |h| < 6}.

These sets were first introduced by A. Denjoy. He showed that with respect to these
sets for 0 S l < i, i = 1,...,k — 1, f.- is an l-th Peano derivative of f,-_1, with
( f1|p6)(,-_,)(2:) = f.-(.'c) for at 6 P5, where the (l — i)-th Peano derivative is computed
relative to P5. Using different techniques, we are able to improve his result. Namely,
we show that the result also holds for the case i = 1:. Since Ufi°=1 P” = R, we have that
fl, is a composite derivative of fk_1. This gives a positive answer to a question raised
by C. Weil. (See [19].) From this result it is easy to conclude that f]. E [A’]. We just
need to recall the fact that for any function g defined on a closed set A, such that
at every point a: E A, a derivative g'(:r) computed relative to A exists, there exists a
function G differentiable at every point a: E R so that 0],, = g and G’ I A = g’.

We can enlarge the sets PM so that they are perfect and that f], is still a composite
derivative of fk_1 with respect to these perfect sets. Therefore one more property
possessed by approximate derivatives is also possessed by Peano derivatives.

Because every composite derivative is a Baire 1 function, we see that f], is a Baire
1 function. Although this property is very easy to establish for Peano derivatives, for
generalized Peano derivatives it is not so easy. We will discuss these derivatives in
Chapter IV, but using some techniques similar to those that we use for Peano deriva-
tives we will prove that generalized Peano derivatives are also composite derivatives
and therefore, they are Baire 1 functions.

Another immediate corollary is that f], is the approximate derivative of fk_1 almost
everywhere. This result was first proved by Zygmund and Marcinkiewicz. (See [20]
page 75.) We will generalize their result showing that fk is the l-th approximate
Peano derivative of fk_1 with (fk-;)(a,,_;)(:r) = fk(:c) almost everywhere, where (ap—l)
denotes l-th approximate Peano derivative.

The sequence of sets {Pu} satisfies the condition Pn C PM]. This fact together
with results already established will enable us to construct a system of paths {Ex : a: E
R} satisfying the LC. property (as it was defined in [3]), so that f], is a path derivative

 

 

 

 

 

of fic—
forpa
prope
(E, :
ta far
and (

Fi
condi‘

that s

Defin
called

wists,

sclcrti

of f1,_1 with respect to this system. Using an induction argument and known results
for path derivatives with such a system of paths, we get that Peano derivatives possess
properties (B), (C), (D), (E), and (G). Since there is a nonporous system of paths
{E3 : x E R}, such that f), is the path derivative of fk_1 with respect to that system,
(a fact established in [3]), we get that Peano derivatives possess also properties (F)
and (H).

Finally we show that there is a system of paths {Ex : x E R} satisfying the I.I.C.
condition (as it is defined in [3]), with fk the path derivative of fk-1 with respect to

that system. This implies that f;‘ is the selective derivative of fk_1.

Definition 0.0.5 If for a given function F there is a function p of two variables

called a selection, satisfying p(x,y) = p(y,x) and p(x,y) E (x,y), so that

. F(p(x,y))-F(Iv)
Iii-'12 p(x.y)-$ (2)

exists, we say that F is selectively differentiable at x, and the limit in (2) we call the

 

selective derivative ofF' at the point x and denote it by F;(x).

Selective differentiation was introduced by R. O’Malley. Motivation for introduc-
ing selective differentiation was the fact that approximate derivatives are selective
derivatives, which was proved by O’Malley. Showing that fk is a selective derivative
of fk_1 we give a positive answer to a question raised by C. Weil. (See [19].) So Peano
derivatives possess one more property possessed by approximate derivatives.

Generalized Peano derivatives were introduced by C. Lee. (See [9].) He showed
that every absolute Peano derivative on a compact interval is a generalized Peano

derivative. Absolute Peano derivatives were introduced by M. Laczkovich. (See [7].)

Definition 0.0.6 Let f be defined in a neighborhood of x. We say that the absolute
Peano derivative off at x exists and is A (in symbols f‘(x) = A) if there is a function

g, a nonnegative integer k, and a 6 > 0 such that

(i) g, = f on (x — 6,2: + 6) and (ii) gk+1(x) = A.

,.
bk.)
'

.f' "will

 

 

 

 

 

wher

II
indefi
that 1
qtold
any t'
same.
Note t
the C0
d9fl\'at

C. .
Wells
they art

We .
that (a;
p(98110 r}
al50 for

PPanO (A

 

Laczkovich showed that this concept is unambiguously defined, that if f “ exists
on an interval, it is a function of Baire class one, it has the Darboux property, and if

f‘“ is bounded above or below on an interval, then f“ = f’ on that interval.

Definition 0.0.7 Let F be a continuous function defined on R, and let it E N. We
say that F is n-th generalized Peano difierentiable at x E R, if there is a positive
integer q, and coefficients F[,](x), i = 1, . . . ,n such that

0'1 (-a +1) F
F(-q)(x + h) =2: h1____ F 11(3) +2 hq+j(q 814:5! + h0+n¢£fln(x, h) (3)
1:0 j=0

where limhno eqq+n(x, h): 0.

Here F[o](x) = F(x) = F(°)(x) and F(‘j)(x) = fI F('j+1)(t) dt; i.e. F(‘jl is an
indefinite Riemann integral of the continuous function F (‘j“) for j = 1,. . . ,q. Note
that the definitions of F[,-](x), i = 0,1, . . . , n and of 65141:, h) don’t depend on which
q-fold indefinite Riemann integral F ('9) of the continuous function F, is taken because
any two differ by a polynomial of a degree less than q. The above definition is the
same as the definition of (q+n)-th Peano derivative of a function F (‘9) at the point x.
Note that every n-th Peano derivative is also a n-th generalized Peano derivative, but
the converse is not true. Namely M. Laczkovich has constructed an absolute Peano
derivative on an interval which is not an ordinary Peano derivative of any order.

C. Lee showed that all properties (A), (B), (C), (D), (E), (F), (G), (H) and
Weil’s Z property are possessed by generalized Peano derivatives and in particular
they are possessed by absolute Peano derivatives. (See [8] and [9].)

We will take a different approach to studying generalized Peano derivatives than
that taken by C. Lee. Our approach will be similar to the one we used in studying
Peano derivatives, so many results that we established for Peano derivatives will hold
also for generalized Peano derivatives. In particular we will obtain that generalized

Peano derivatives are composite derivatives and hence belong to [A’]. Also we will

show that generalized Peano derivatives are path derivatives with respect to a bi-
lateral, nonporous system of paths satisfying I.I.C. condition. Therefore generalized

Peano derivatives are also selective derivatives.

show that generalized Peano derivatives are path derivatives with respect to a bi-
lateral, nonporous system of paths satisfying I.I.C. condition. Therefore generalized

Peano derivatives are also selective derivatives.

 

 

 

Thrc
on R. B?

1.1 I

Let C l)
functior
defined
prOpertl
f = I».

four (01

Th.

Corlilm
prOVe ,

deriya,

CHAPTER I

Throughout this theses all the properties will be established for functions defined

on R. But it can be easily seen that R can be replaced by any connected subset of R.

1.1 Decomposition of Peano derivatives

Let C be the family of all continuous functions on R, A the family of all differentiable
functions on R and A’ the family of all derivatives on R. If I‘ is a family of functions
defined on R, then by [I‘] we denote the family of all functions f on R with the
property that for every n E N there exist v7, 6 I‘ and a closed set A, such that
f = vn on An and U;,'°=1An = R. In [1] (Theorem 2) it is shown that the following

four conditions are equivalent:

(i) There are g, h and k in A such that h’, k’ E [C] and f = g’ + hlc’.
(ii) There is a cp E A’ and 2/) E [C] such that f = cp + 11).
(iii) The function f E [A’].

(iv) There is a dense open set T such that f is a derivative on T and f is a derivative

on R \ T with respect to R \ T.

The statement (ii) implies that [A’] is the vector space generated by A’ and [C].
In [1] (Theorem 3) it is shown that each approximate derivative, each approximately
continuous function and each function in Bf = [C] belongs to the class [A’]. The
main goal of this chapter is to show that every Peano derivative is in [A’]. We will
prove even more. Namely we will prove that every Peano derivative is a composite

derivative.

10

11

Definition 1.1.1 Let f be a function defined on R. If there exist a function g, and
for n 6 N there is a closed set An with gI’An(x) = f(x) V2: 6 An and Ug‘fflAn = R,

then we say that f is the composite derivative of g.

The following result due to Maiik says that every composite derivative is in [A’].

(See [14].)

Theorem 1.1.2 Let a function g be defined on a closed set H. If 9’ exists on H,
where g’ is computed relative to H, then there is a function G differentiable on R so

that CI" = g and C’IH = g’.

O’Malley proved that every approximate derivative is the composite derivative of
its primitive. (See [11].) In this chapter we will prove that a k-th Peano derivative
is the composite derivative of the (k — 1)-th Peano derivative. Thus we will get that
every Peano derivative is in [A’], and hence possesses all the properties possessed by

functions in [A’]. We will start with an elementary lemma.

Lemma 1.1.3 For m E N we have

0 ifi=0,...,m—l
m! ifi=m

%(m+l)! ifi=m+1.

f:(-1)"‘"'(';‘)j‘ =

i=0

Proof: Let Bf" = m (—1)"“‘j(’;.‘)j’. Then B? = 0, B11 = 1, and B? = 1.

i=0

Now we will proceed by induction on m. Suppose

{ 0 ift=0,...,m—2
Bid:

m

(m-l)! ift=m—l
m2:_1mg ift=m.

Note that B3, = (1 — 1)"‘ =0. Let 1 gig m+1. SinceiZ 1, 0’ =0. Thus

Bf. = mi(—1)’”‘j(';‘:,‘)j"‘

.=1

12

111-1

= m §(-1)"‘“" (m;‘)(j + 1)"-1

m-l i—l

= m 2(—1)m-1—1(mj—1)): (i:1)J-i-1-r

j=0 r=0

i-l
= m 2 (':1)B;,,'lf' , and by the induction hypothesis
r=0
0 if i — 1 < m — 1
- mB,',",:] ifi—1=m-l
m((T)B,',’,‘:11+('g)B,T,_1) ifi _ 1 = m
0 ifi < m
= m(m — 1)! if i = m
m(m(m — l)! + 951ml) ifi = m +1
0 ifi=0,...,m—1
ml ifi = m
Lg-(m+1)! ifi=m+1. Cl
Definition 1.1.4 The Riemann diflerence A?f(x) of order m ofa real valued func-

tion f at a point x is defined by Arflz) = 2;":0(—1)’"‘j (T)f(x +jt).

If f is continuous on R, then Atf(y) is continuous on R. This is the case if the
k-th Peano derivative f1, exists on R.

The relationship between A? and A?“ is given by the following simple lemma.

Lemma 1.1.5

AI""‘f(fl=) = Al‘fla? + t) - ANGIE)-

Proof:

Arftz + t) - AI"f(-’v) =
Z(_l,m-.(?),(, +(j+1)t)— Z(-—1)"‘"'(';-‘)f(x + it) =

i=0 i=0

f(x + (m +1)t)+ i(—1)m+1-i( :31) + (3‘)) f(x + jt) +

i=1

 

 

for ti
1. Ila in

Leninia 1

Emma

Then for

 

Hmf

A?“

 

TM

 

TilEOr
Then ..

m.

with n

 

l3

(—1)"‘+1f(x) (since (if!) +(';.‘)= ("‘1“) it is equal to)
m+l . .
= §(-1)m+“’(mf‘)f(x +Jt) = Arenas)- :1
J:
For the remainder of this chapter I: will be a fixed positive integer greater than
1. If a function f at some point x E R has a k-th Peano derivative fk(x), then using

Lemma 1.1.3, we have the following formula for Riemman differences.

Lemma 1.1.6 Let f be a function defined on R. Let x E R be such that fk(x) exists.
ThenforOS m S k

 

t"zjzot—1)m-j(';)j‘e.(z,jt) ifm>i-
Proof:
Arftx) = :(—1)m-J'(’;‘)f(x+jt)
= fan) 1) MC; )(§;:O(j) t)"——’(;”+ I) +(jt)e.(x,jt))
= :t'flif) §(-1)m”(,)j’ +t‘Z(-1)"‘ 1(7')j 6,‘($, jt)

and by Lemma 1.1.3 we have
tmfmtx) + ,. 2,-.(-1)m-J( ),... emtz jt) if m— _.-
t'E,"f_.0(—l)m’1(';.‘)je,-(x, jt) ifm>i. C1

The next theorem is an easy consequence of Lemma 1.1.6.

Theorem 1.1.7 Let f be a function defined on R such that fk exists for each x E R.

Then f), is a Baire 1 function.

Proof: For each n 6 N let g..(x) = n"A" %.f(x) Then by Lemma 1.1.6 (applied

with m = i = k) we have lim.._.oo g..(x) = fk(x). It remains only to notice that each

14

gn is a continuous function. Therefore fl; is a pointwise limit of continuous functions;

i.e., ft is a Baire 1 function. E]

In order to prove that every Peano derivative is a composite derivative, we need
to construct a sequence of closed sets {An} whose union is R, and with respect to
which f], is a composite derivative. The obvious candidate for a primitive is fk_1.
We will investigate a relationship among the Peano derivatives f,- for i = 1, . . . , k on

certain sets Fk( f, e, 6) which are defined next.

Definition 1.1.8 Let c > 0 and 6 > 0 be given. Let

P = Pk(f,e,6) = {x : [ek(x,h)| < 6 whenever |h| < 6}.

These sets were first introduced by A. Denjoy. (See [5].) He proved that for
i = 1,. . . , k—l and for 0 S I g i, f,- is the (i—l)-th Peano derivative of f; with respect
to the closure of these sets with the expected values; i.e., with (fIIF)(,-_1)(x) = f,-(x)
for every x E F. (He proved that the same conclusion holds if f is defined on some
perfect set H having finite index provided fk, computed relative to this set, exists
on H.) (See [5].) When f is defined on R so that fk exists on R (or f), exists on
a perfect set H), we will show that the result also holds for i = Is. First, using
different techniques than Denjoy used, we will prove his result. We begin with some

elementary formulas.

Formula 1.1.9 Let f be a function on R and let x E R. Forl E N, suppose that
fz(x) exists. Then for each t E R

£1-1(x, t) 2 Ag?)- + te;(x, t).

15

Proof: The assertion follows directly from the definition of Peano derivatives. II]

Formula 1.1.10 Let f be a function on R and let x,t E R. Forl E N, suppose that
fz(x) and f;(x +t) exist. Then

f:_1(:c+ )— fz1(z)—tfz(x)=tZ(—1)’j(,’)j’q(:v,jt)+

i=1

l-l
;(_1)l—l-j (l;l)jl—l€l_l(x,jt)_

1

(—1)"1—’(’j1)j"1q-1(x+t,jt).

1

~
I

J

Proof:

By Lemma 1.1.5 A[f(x) = Ai’1f(x + t) — A[‘1f(x).

Applying Lemma 1.1.6 to both sides of above equality we get
1

t’ f;(x) + t’ 2(—1)'-J‘ (9.4.1., jt) =

j=1

tl— lfl— 1($+t) )-—+tl 11-21(_ 1)l—1—j(l— 1)J’l-l Cl— 1(1: +t,jt)—

:0
Wm. wa— 1)-"-J( )-'-I..-.<z,jt>.

Dividing both sides of the above equality by t"1 gives the desired formula. [:1

Formula 1.1.11 Let f be a function on R and let x,t E R. Forl E N, suppose that
fl($) and f1(:c + t) exist. Then

1+1

fz(:c+)- f:(=x) Z(— 1'+‘-J('+‘)J 61(rv 20+

j=0

1 1
Z(-1)l-j(,i)jléz($ajt) — EI—lll-’ (Djlfle‘ + t1jt)‘

i=0 i=0

16
Proof:

By Lemma 1.1.5 A[+If(x) = A[f(x + t) — A[f(x).

Applying Lemma 1.1.6 to both sides of the above equality we get
1+1 _

t' Zt—IWH(‘:‘)j'e.(x,jt) =

l
was +t) +t'zt—1)’-j(;)j'e.(z + wt) —
I
tlMil?) —t12(—1)"’C)1161($,fl)o
j=0

Dividing both sides of the above equality by t’ gives the desired formula. C]

Formula 1.1.12 Let f be a function on R and let x,t E R. For I E N, suppose that
f1(x) and f1(x + t) exist. Then

fz-1($ + t) - ftlxl
t

 

l
41x)=¥(—1)’*’G)i’a<evit>+

l-l
’—;l(f.(x) - fz(ar + t» + gnarl-1' (fireman) — .,(. + tut».

Proof: From Formulas 1.1.10 and 1.1.9 we get

fz-1(w+t)—fz(z) Hm )=tz(- 1)::C),z.,(.,,-.)+
l-l
g<—1>'-“"(‘;‘)j'-‘(jt, tff" —)+ +1:th jt))-

E(-1)"""(’§‘)i"1(’zf"‘(—."‘$ 1+ ) +Jt61(1‘ +t jt))

Dividing both sides of the above equality by t, and applying Lemma 1.1.3 we get the

desired formula. [3

17

Theorem 1.1.13 Let f be a function defined on R such that f), exists for each x E R.
There is a positive constant M such that V e > 0 and 6 > 0 if x and y are in P with

Ix — y] < 1%, then

 

lfk(y) - M3” 5 M‘ (1)
fk-lIyz : ilk-1”) — fk($) 5 Me. (2)

Moreover for [h] < 6 l: 1,. . . ,k f; and e;( ~ ,h) are bounded on P 0 [a,b] inde-
pendent of h, for any interval [a, b].

Proof: Let x and y be in P such that |y — x] < ti], and let t = y — x. Set
B = 2:3 ( k j 1 ) j". Then the left hand sides of the equalities in Formulas 1.1.11
and 1.1.12 are bounded by 3B6 and 3B6 + 3Beig—1 respectively. Hence (1) and (2)
follows for M = 3B 5-125-

Let [a, b] be any interval. From (1) we see that ft is bounded on P n [a, b]. From
Formula 1.1.9 (applied with l = 11:) it follows that for |h| < 6 Iek_1( - , h)] is bounded
on P n [a, b] independent of h. Now from Formula 1.1.10 (applied with l = k) we see
that fk_1 is bounded on P (1 [a,b], and again going back to Formula 1.1.9 (applied
with l = k— 1) we see that for Ihl < 6 ]6k_2( - , h)| is bounded on Pfl[a, b] independent
of h. Continuing we can deduce that there is a constant C so that I f;(y)] S C and

|ez(y,h)| S C whenever y E P 0 [a,b], |h| < 6 and 1 _<_ I S k. C]

The next theorem says that if we replace P by P, then the conclusion of Theo-

rem 1.1.13 still holds.

Theorem 1.1.14 Let f be a function defined on R such that f]. exists for each x E R.
Then P Q P(f,3e,6).

18

Proof: Let x E P, and let {xn} E P be a sequence such that limn...» xn = x. Let
[a, b] be such that {xn} C Pfl[a, b]. From Theorem 1.1.2 we see that f; for 1 S l S k
is bounded on P n [a, b]. Therefore we can choose a subsequence {xnj} converging to
x such that for i = 1- - - k the sequence {f,-(x,,j)} is convergent. Let these sequences
converge to F,-(x), i = 1,... , 1: respectively.

Let h with [h] < 6 be given. Suppose that [h + x — 3.1,] < 6 for everyj E N.
Therefore Iek(x,,j , h+x-xn,)| < 6 so we can also suppose that this sequence converges.

(If not, then extract a convergent subsequence.) Denote its limit by E(h). Since

fk(x,,j) exists,

(h + x — x,,1.)"‘1

 

f(x + h) : f(xn,) + (h +37 "' $n,)fl(xn,) + + fit—10311,)

 

(k — l)!
+ (h + x — xnj)"_l(h + x — xn1)(fk(:!n’) + ek(xn’., h + x — 33%)) . (3)
Lettingj —) oo in (3) we get
k—l x
f(x + h) = f(x) + hF1(x) + - -- + MIT’s-1(3) + h""1 h (5% + E(h)).

Since limh_.o Milk?) + E (h)) = 0, by the uniqueness of Peano derivatives we have

E-(x) = f,-(x) for 1 S i S 19—1 and

 

 

fig) + an, h) = Eff) + E(h). (4)

Since |E(h)| S e, from (4) we have that

fk(z) - F.(::)
k!

 

= |E(h) - 61(x,h)| S 6 + l€k(-’v,h)|- (5)

 

 

The left hand side of (5) doesn’t depend on h so letting h —+ 0 in the right hand

side of (5) we get MW

Finally this estimate and the formula ek(x, h) = E(h)+m9fi’—*Jfl gives lek(x, h)| _<_

_<_e.

 

 

26 < 36 for [h] < 6. Hence x E P(f, 3e, 6) and the theorem is proved. Cl

19

In order to prove that Theorem 1.1.7 holds for i = k, we need a formula that
involves more than two variables. We will derive a formula, (Theorem 1.1.17 below)
involving three independent variables. The proof of the formula is elementary, but

the formula itself is the crux in what follows.

Lemma 1.1.15 Let 0 S s S k — 1, x1 E R and let a function f be defined on R,

having a k-th Peano derivative at x1. Then

I: s
Aif($l) = t'fa($1) + 2 Ex- 1)‘-J(;j) lt___ flu-"(31)

l=s+1j=0

tk Z(-1)"’(,-)J €k($1,Jt)

Proof:
.1 I:
Am.) = gem-1L) (it ‘(It)‘f—'——‘)+(Jt) 6.6. m)
a k
= gt—w- (,);:‘t"’(x‘)+t"2( wet-k 6:.(x1 m
k 3 s- a .llltf—J—(fl) I: .—
= ;;(-1) 101 +t ;(— 1) J(;)j 6.61.11)

by Lemma 1.1. 3

= zit- w-Jc)'1‘—,——f'<s)+tk:’:< 1)'--'(;)j* at...»

1:3 j==0 j=0
Applying Lemma 1. 1. 3 once more

Ami) = tf.(:c1)+ z; :(_1)--J(;)I£__f'($1)+

l=s+l J=0

”EX- 1)“’(,-)J'" 61061 M n

j=0

The next lemma gives a different formula for A: f (x1) when we suppose that the

k-th Peano derivative of a function f on R, exists at a point x gé x1.

20

Lemma 1.1.16 Let 0 S s S k— 1, x1 and :1: two points in R. Suppose that a function

f defined on R, has a k-th Peano derivative at x. Then

Proof:

Aim:

1)

k
Aff($1) = [(31

)l—sts fl($)

_ 1- ___+
l=s (1 — 3)!

Z X (i)1(- x)"'t'z:(_ _1)8-J'(;)jc f_l_(-’€)+

l=s+l i=s+l

Z("1)8_j(;)(1'1 — $+jt)k 61¢(x, $1 —:c+jt)

:I—1)’—’ (3) E(xl - x + jt)’-f——l(!x+ x)
l

1:0 . l=0
;(—1)"°’(; (x1 — x +jt)"ek(x,x1 — x +jt)

I: I
2;:(—1)s-j(:) g (2 (3(31- Ivy-ijitH f_l__($ )+
git-1) -1. (5)051 — 1’3 +Jt)k €k($, $1 — a: +jt)
k I I Hi ' s—j s -.' It”)
Z; (Jul - .) t (got-1) (J). ) T +

:(-1)“j(;)(x1 - x + jt)"ek(x, x1 — x + jt)

i=0

which by Lemma 1. 1. 3

ZZ(.-)($1)"‘t' (Z(—1)"j(3)j‘);f__’(!x_(_)

l=s i=5

:(-1)a-2(;)(.,_ . + Watts: .1 — . + jt)

j=0

Elia-Macks): ”(25(- WC) )f_z__(x)

(=8 j—O

20

Lemma 1.1.16 Let 0 S s S k— 1, x1 and x two points in R. Suppose that a function
f defined on R, has a k-th Peano derivative at x. Then

Proof:

Ail-($1)

k
A?f(:v1) = [(31

)I-ats f1($)

1:, — 1' (I - 8)!+

2 5'; (2m) ,...,.,«3( 1)’—J(;) I’m—p

I=a+l i=s+1 J'__0

Z(—1)"j (;)(x1 — x + jt)kek(x,x1 —— x +jt).

i=0 (:0

:14)” (,)($1 — a: + jt)"ek(x, 3:1 — x + jt)
3 k I 23
§(_ _1) -J' (2) E; (g (3(31— x)l'i] iti)__l_!__ f_l___( )+
§(—1)"’(,~)(x1 - x + 10 6k(x x1 — a: + jt)

2323(’.-—)(x1 x)" w (Z(—1)""(3) )f__z(x)+

I=Oi

23(-1)”’(§)(=v1 - I + fl)" 6:.(x, 2:1 — a: + jt)

i=0

which by Lemma 1.1.3

2:21: (1)031 — x)""t" (:(_1)3-J 03—17—43) fz_(__rL‘)+

(:8 i=8 j=0

i:(—1)’—j(;)(x1 — a: + jt)*e.(:c,x. -- x + jt)

j —0

2300131 — :L‘)""t‘ (23-1)»: (1) ._T_)+fz_($)+

(=3 j=0

21

2 2 (1.x..- x)"‘t'2(_1o-J’G) f__z(x)+

l=s+l i=s+1
gem-icy...—.+.1r..(.,.._.+...

applying Lemma 1.1.3 once more yields
k l I 3!
A:f(a:1) = )3 (,)(x. — .) “rims +

(:3

k 1 I 1-. _ ,-f_z(x)
Z .2 (W1 -..) 't‘2(— 1) ’6)!" +
=s+1i=s+1
:(_1)'-j(3)(31 - 95 + fired-r. m1 — :1: + jt) =
" -.. f(x)
.231”le t (IL—T)! +

k )‘l- _ .f___z($)
2.2.0114 12-12.) +

E(— 1)”’ (J)( (x; — x +jt)"ek(x, x1 — x +jt). D

Putting Lemmas 1.1.15 and 1.1.16 together we get a formula that is the crux of
the proof of Theorems 1.1.20 and 1.1.27.

Theorem 1.1.17 Let 0 S s S k — 1, 0 74 t and x1 and x two points in R. Suppose
that for a function f defined on R fk(x1) and fk(x) exist. Then

($1 -$)"’
f.(x1)- Z (l— 8,) ———-=fz($)
[=0

2(-1)“"(3) 2i. t"’(2(”“ _f):————:fl(x)— f.(a=.))+

j=0 i=a+l

 

§(_1).—j(1)(z1 — x + jt)"£k($, 1’1 _ x + j,)_

t8

22

1'“2(—1)’-J'(;)J*e.(x.,J1).

J'=o

Proof: By Lemma 1.1.15 and Lemma 1.1.16

fa($1)— 2(ngL—‘ $1-.) fl($)=

 

(=3 _‘9!)

k I x)l-i i- -s —j if“: _)_+
l§+1£§1(W)( t 12:;— 1) (3)].
g(_)_1)a-j(j) (31- :‘+Jt)k;($,$l —$+jt)—
Z Z(_1)a—j(:)jsltl—a;fl( 1___)__
l=s+lj=0

t"“ 2(— 1)"" (J)j" ek(x1, jt) (changing the order of summation)

j=0

=Z(_1)-J()) Z Z(f)(x1x)"“t"j‘f—'—($)—

i=s+l l=i

s k , $1
Z(_1)s-J(;) Z jiti-sf'_(i!_)+

j=0 i=s+l

’ - x1 —x ' k .
E(-1)’_J(§)( Ht) “(£31 - 1*? +Jt) -

to

Jk-o;<- 11-10. .1... J1)
= 2(-1)‘*"(3) 2 3v“ (2 (fi) (.1 "JV-“11(1) — mm) +

j=0 i=s+1 ' 1::

 

 

 

§(_) _1)a—j(;)($1- Itc‘+jt)k€k($,x1_$+jt)_

t"“23(— 1)” (,)Je1(:c1 J't)

i=0

=2(—1)’-’(;) 2,. ’— t"(2”‘(,'_ -—-:-:fz(x)— f.(xl))+

i=s+lz l=i

23

 

, $1 —27 . k .
§(_1)s-j C)( + Jt) 61437,3151 — II? + N)—

t8

t"” 2(— 1)‘ j(; )jk ek(:c1,jt).D
The following formula is the special case of Theorem 1.1.17 where t = $1 — :13.

Formula 1.1.18 Under the assumptions of Theorem 1.1.17 the following formula
holds:

,().’B1 “'2 (____$(II:$))I!— f((x) :

l=s 3)

a l: _—xl —i
2(—1)'-"(;)21i.—.(x1—x)-'(2(———’(‘,_.§ fz(x)—f.-(x1))+

i=s+ l-i

2(-1)*-J'(;)(a=1- .)k—s(1+.)k..(.,(.. — x)(1 +1»-

i=0

($1- 1') )k" ’2]— 1) ’ ’(j)j 6:.(z1 1(31 -$))

The next theorem is a generalization of Theorem 1.1.13.

Theorem 1.1.19 Let :c,:c1 6 F(f,c,6) with [2:1 —:c| < fif’ Then there is a constant

M not depending on e or 6 such that

k -a
f,($1) _ 2 ($1 — x),

Wfih’) _<_ '1'1- xlk—'Mc
1:, .

fors=l,...,k.

24

Proof: The proof is by induction on s. The case 3 = lc follows from Theo-
rems 1.1.13 and 1.1.14. Suppose it is true for all s + 1 S i S k. Therefore there is a

constant M1 so that

$1_ l—i
f.(x1)—2(——(,_f) 11(2) <Ix1—xl*‘Mle (6)

l=i

for i = s + 1,. .. ,k. Then Formula 1.1.18, Theorem 1.1.14 and the induction

 

 

hypothesis (6) yield

($1-$)'”
fa($1)- 12:2“ _3!) ——_fl(1‘)

:(j)z _lxl —a:|' ’le —x|k 'M16+

 

 

j=0 i=s+lz
:C)|x1 _ a:|"”(1 +j)k3e + (since F(f,e,6) c P(f,3e,6))
lwn — xI’H 2 (;)j*3e (since 75(f. «6) c F(f.36.5))

= [2:1 — xlk"Me

where M is a constant that does not depend on c or 6. The induction is complete as
is the proof. Cl

Now we are ready to prove our main theorem.

Theorem 1.1.20 Let f be a function defined on R such that f). exists for each a: E R.
Suppose 0 < s < k and 6 > 0. Then f, is (k — 3) times Peano diflerentiable with
respect to PU, 1,6) with the expected value; i.e., (f,|-,-,-)k_,(:c) = fk(:c).

Proof: Let a: 6 F(f, 1,6) and let 1 > e > 0 be given. Then there is a0 < 17 < 6such
that |ek(a:, h)| < 6 whenever |h| < 17. Let M be the constant from Theorem 1.1.19. Let
2:1 6 P(f,1,6) so that Ian —:r| < 7:313. Let t- — (ml—flei. Then forj = 0,1,...,k—1

we have

25

1:1 — a: +jt|= (1+je%)|:r1 — :cl < k|xl — xl < 17. Hence (7)
lek(:r,a:1 — a: +jt)| _<_ c and (8)

|€k(rt1,jt)| S 3- (Since-17f. 1.5) C P(f.3.5) and kltl _<_ klxl — 36| < n < 61) (9)

By Theorem 1.1.17

I: ($1 _ 3))l!-s
f,($1)— ”—2 (l _ 3)f1($) =

l: s

a k '3' $1 _ it l-i
E(—1)“"(§) Z 27‘” (2 fifth) —fi($1)) +

 

 

 

j=0 i=s+1 2! l=i

' . , — + 't " .

§(_1)8—J (j) ((131 :9 J ) €k($a$1 _ $+]t) _

tk” :(— 1)‘ ‘1 (Djk £k(x1,jt) and by Theorem 1.1.19
j=0

 

3 (ah-.11-.) 1?
f.(1)2—1———(1_f:())

s k
2 C): —|:rl — :1:|' ’eTMle — xlk‘ "‘

j=0 i=s+lz

s a -8 1+ ‘él/lc k .
2(j)|x1_$lk ( :13; )|€k($,$1—$:Jt)|+

 

 

 

j=0
Ix. — sunk-2‘? 2 (;)j"lc.(z1,jt)| and by (8) and (9)
j=0
8 k 'i . a - k
S E (3) 2 lel -— :rlk”c"E£M + 2 (film — :cl:”-(ii.ll—e +
j=0 i=s-lv-1z j=0 (I
|x1- w- 2‘14 2 (;) *3- — le new): (9f; 2%? +
1:0 j=0 i=s+1z
lml — :rIk-’ 1‘: ()(1 +j)ch + |a:1 — xl" ’3: C)j" 5T
1:0 j=0

Since 6 was arbitrary, we have that

hm f.$121_.L17—)yI-n:mfz($)_o.

1'1 €P.$1-'Z ($1 " 37),“.

26

Therefore the assertion of the theorem is proved. C]

Theorem 1.1.20, together with the simple observation that U°° Pk(f,1,1/n) = R

n=1

has many applications.

Corollary 1.1.21 Let f be a function defined on R such that fk exists for each x E R.

Then f), is the composite derivative of fk_1.

Definition 1.1.22 A function f is said to have a k-th approximate Peano derivative
at x if there exist numbers fap(1)(x), fap(2)(x), . . . ,fap(k)(x) and a set V with density
1 at 0, such that
hi:
f(x + h) = f(x) + hfauo(1)(17) + ' ° ° + k—!fap(k)($) + “(13: h) (10)
where ek(x, h) —+ 0 as h E V, h —» 0.

For k = 1 we have the definition of the approximate derivative.

Corollary 1.1.23 Let f be a function defined on R such that fk exists for each x E R.
Then f, is almost everywhere (I: — 3) times approximately Peano differentiable with

the expected values; i.e., (f,)a,,(k_,)(x) = fk(x) for s = 1, . . . , k - 1.

Proof: Let x be a point of density of P = Pk(f,l,% . By Theorem 1.1.20,
(f,|p)k_.($) = fk(x). Since x is a point of density of P, we see that (f,)a,,(k_,)
exists at x and equals fk(x). Finally the Lebesgue Density Theorem and the fact
that U$,°=1Pk( f, 1, i) = R proves the corollary. D

The case .9 = k — l was proved by Zygmund and Marcinkiewicz. (See [20], page

77.) Corollary 1.1.23 can be regarded as a generalization of their result.

27

Corollary 1.1.24 Let f be a function defined on R such that fk exists for each x E R.
Then f1, 6 [A’], and hence

(i) there are g, h and q in A such that h’,q’ E [C] and fk = g' + hq’,
(ii) there is a go 6 A’ and 11) E [C] such that fk = so + w,

(iii) there is a dense open set T such that fk is a derivative on T and fl. is a derivative

on R \ T with respect to R \ T.

We will end this chapter with a different decomposition of R into closed sets so

that fl. is the composite derivative of fk_1.

Definition 1.1.25 Let f be a function defined on R such that f], exists for each x E R
and let

H(f.M.6)={x= “1

 

k-l
no) + 2(-1)"“"’(*E‘)j*e.(a=.jt) s M for M < 6}

where M and 6 are positive constants.

Lemma 1.1.26 Let f be a function defined on R such that fk exists for each x E R.
Then for any 6 > 0 we have U°fi=,H(f, M, 6) = R.

Proof: The assertion follows from the fact that 6;.(x, jt) is a continuous function

oftforj=0,1,...,lc-1.D

Theorem 1.1.27 Let f be a function defined on R such that f}, exists for each x E R.
Then H = H ( f, M, 6) is closed and fk-1 is differentiable on H relative to H with

fk_1|’,,(x) = fk(x), also Ifk(x)| 5 2M for every x E H.

28

Proof: Let x E H. Let 1 > e > 0 be given. There is 0 < 17 < 6 such that
l€k($,h)l < 6 whenever |h| < 17. Let xn E H so that Ixfl — xI < 53-. Then for
t=(xn—x)e% wehaveltl <6and |x,,—x+jt| <n,forj =0,1,...,k—1. By the

formula from Theorem 1.1.17 applied with s = k — 1

lc-l
fk—1($n) - fk—1(-’I?) - (17a — $)fk($) = t§(—1)k—l—j (k;1)fk(1‘) +

 

k—l . k

_ _ ' _ x" _ $ + t .
Z(-1)* ‘ 1(2‘)‘ W J ) ems—Hm—
i=0

k—l

t):(—1)*-‘-J (*g‘)j‘=(f.(x.) + cummjt».
i=0

By Lemma 1.1.3 we have

fk-1($n) — fit-1(3) _ fk($) = x t 3k ; 1fk($)+

inn—3 n“

 

 

E(_l)k—1—j (1:71) (“in - 93 +1.0]:

tk-1(xn—x) ek(x,xn—x+jt)— (11)

i=0

t

(En—23 (kglfk (3n )+Z(— 1) k- -1—j(k; l)jk (“37:11.”)

:3 = 6%, from (11) we have

 

       

fk—1($n) - fie—1U?)

xn—x

e1

 

- fk(1')S

 

llfk(-T)l +

 

 

I: 1 t
(k; I)(———'—— ltje ) |€k( 1” xn-x+Jt)l+
i=0 CT

 

(12)

 

k—l *4 ,_,_
f.(x.)+2(-1) J(*; )je.(x.,jt)

Since Ick(x,x,, — x +jt)| < c and since xn 6 H(f, M, 6), the left hand side of (12)

 

is

k—

 

lIf:.(x) |+ Z (*;1)(1 +jcr)"er + erM.

j=0

29

Therefore if {xn} is a sequence in H converging to x, then

fir—1(zn) fk—1($)

mn—

-fk($)—*0-

 

Let x 6 H, {xn} a sequence in H so that x" —-> x and let 0 # |t| < 6. Then (11)

 

 

 

 

 

 

 

yields
_ k—l _ x _ z ' k
t (k 2 1f.(.)+ 2(—1)"-“’(";‘)( " .5") mm — 2: +jt)) s
t (k _ 1fk(xn) + E(“Uk-l-j (k71)jk€k(xn .7”) +
2 i=0 1 ’
|fk-1($n) - fk_1($) — (17a - a¢)fk(-”B)| S
lth + |fk-1($n) - fk_1(:v) - (3n - 1‘)fk($)|- (13)

Letting n -—+ co the left hand side of (13) becomes

 

t k‘lf k-l lk-l—j k—l -k 't
2 .(x)+2(— ) (, )2 “(an

 

 

while the right hand side of (13) is Ith. Hence a: E H.
That I fk(x)| 3 2M on H follows from Definition 1.1.25 taking t = 0. E]

Theorem 1.1.27 and Lemma 1.1.26 combine to say that fk is the composite deriva-
tive of fk..1, a result that we already established. But this can be regarded as a
simpler proof of that result, because the only tool we used was a special case of The-

orem 1.1.17, whose proof is even more elementary than the proof of Theorem 1.1.17.

On the other hand the sets H ( f, M, 6) are already closed.

CHAPTER II

2.1 Peano and path derivatives

We will start this chapter with the notion of a path derivative that was introduced

in [3].

Definition 2.1.1 Let x 6 R. A path leading to x is a set E, C R such that x 6 E3
and x is a point of accumulation of Ex. A system of paths is a collection E = {Ex :

x E R} such that each Ex is a path leading to x.

Definition 2.1.2 Let F : R —+ R and let E = {Ex : x E R} be a system ofpaths. If

Hm F(y) - F(x)

yeExw-w y — x

 

= f(x)

is finite, then we say that F is E-difierentiable at x and write Fg(x) = f(x). IfF
is E differentiable at every point x, then we say simply that F is E differentiable; we

call F an E-primitive and f an E-derivative.

Definition 2.1.3 Let E = {E3 : x E R} be a system of paths.(IfE has any of these
properties at each point, then we say that E has that property.)
E is said to be bilateral at x if x is a bilateral point of accumulation of Ex.

E is said to be nonporous at x if E: has left and right porosity 0 at x.

The basic definition of porosity of a set E at x from the right (left) is the value
lim sup,_,0+ l(x,r,E)/r , where l(x,r, E) denotes the length of the largest interval
contained in the set (x, x + r) H (R \ E) ((x -— r, x) n (R \ E)). Porosity 0 at x means

both right and left porosity 0. Note that a nonporous system is necessarily bilateral.

30

31

Definition 2.1.4 Let E = {Ex : x E R} be a system of paths. E will be said to
satisfy the condition listed below if there is associated with E a positive function 6 on
R so that whenever O < y — x < min{6(x),6(y)}, the sets E, and Ey intersect in the

stated fashion:

i) intersection condition 1.0.: Ex 0 E3, 0 [x, y] 9i 0;
ii) internal intersection condition I.I.C.: E, D E” 0 (x,y) 95 0;

iii) external intersection condition E.I.C'.:

Eanyfl(y,2y-x)5£0 andEanyfl(2x—y,x)7é0

We will prove that for every k-th Peano derivative fk there is a nonporous bilateral
system of paths E satisfying LC. and I.I.C. conditions, for which f), is the E-derivative
of fk_1. In this chapter we will prove that E satisfies only the LC. condition. To
show this first we will prove the following theorem due to Mafik. (See [13].)

Theorem 2.1.5 Let k E N, x E R. Suppose that a function f has a k-th Peano
derivative at x. Define P(y) = 215:0 y — x)‘—‘},9 (y E R). Let c > 0, n > 0. Then
there is a 6 > 0 such that if I is a subinterval of (x — 6,x + 6), j an integer with
0 < j S k and if either f(j) S P”) on I or f(j) Z P”) on I, then m({y E I :
|f(5)(y) — P(j)(y)| Z er — xlk’j}) _<_ 1] ~ (m(I) + d(x, 1)). (Here m denotes Lebesgue

measure and d(x, I) denotes the distance from x to I.)

In order to prove Theorem 2.1.5 we need two lemmas.

Lemma 2.1.6 Let f be a monotone differentiable function on a bounded interval I .
Let c > 0, [3 > 0 and let m{x 6 I: |f’(x)| Z e} 2 fl. Then there is an interval J C I
such that m(J) = ,6/4 and that |f| 2 63/4 on J.

32

Proof: We may suppose that f’ 2 0 on I. Let (a, b) be the interior of I. There is
3. c6 [a,b] such that f S 0 on (a,c) and f 2 0 on (c,b). Set B = {x E I: f’(x) 2 e}.
If m(Bn (c,b)) 2 fl/2 and if x E (b— fl/4,b), then f(x) 2 f: f’ 2 e-m(Bfl (c,x)) 2
6 ° (m(3 n (c,b)) - (b - 93)) Z 6 - (fl/2 - W4) = 6fl/4- If m(3 n (0,6)) 2 3/2, then,
analogously, f S —cfl/4 on (a,a + 3/4). C]

Lemma 2.1.7 Let I be a bounded interval and let j be a natural number. Let g be
a function such that either gm 2 0 on I or gm 5 O on I. Let c > 0, 5 > 0 and let
m{x E I: IgU)(:c)l Z 6} _>_ 6. Then there is an interval J C I such that m(J) = ,6/4j
and that Igl Z cflj/4‘+"'+j on J.

Proof: The assertion follows by induction from Lemma 2.1.6. C]

Proof of Theorem 2.1.5:
Let g = f — P, and let a = 41+"'+". There is a 6 > 0 such that for each

y E (x—6,x+6) we have
3"alg(y)| S f’lkly- xlk- (1)

Now let I be a subinterval of (x — 6,x + 6) and let j be an integer, 0 < j S k.
Let B = {y E I : |g(j)(y)| Z ely — xlk‘j}, B = %m(B). Suppose that 3 > 0. Let
C = B\ (x — ,B,x + fl). Now Igml Z 6,3(k-j) on C and m(C) 2 B. If either g(j) _>_ 0
on I or g”) S 0 on I, then by Lemma 2.1.7, there is an interval J C I such that
m(J) = ,8/4j and that

'9' Z léflk_j ' flj = 16.3,‘ on J. (2)
a a
Together (1) and (2) yield (3fl)" _<_ nkIy-xl" for every y 6 J. Hence m(B) _<_ 17d(x, I).

C]

33

Definition 2.1.8 A real valued function f defined on an interval I is said to have
the intermediate value property if whenever x1 and x2 are in I, and y is any number
between f(xl) and f((Eg) , there is a number x3 between x1 and x; such that f(x3) = y.

A function having the intermediate value property is called a Darboux function.

It is known that a k-th Peano derivative, fk, is a Darboux function. Also it
is known that if f], is bounded either from above or below, then the k-th ordinary
derivative, f(k), exists with the obvious equality, f“) = fk. In the next theorem we
will only assume that these two properties hold for any l-th Peano derivative where
O S l S k — 1. We know that any continuous function is Darboux, so for k = 1 the

above assumptions trivially hold.

Theorem 2.1.9 Let lc, l E N, with l S k — 1. Assume for each function g defined on
an interval I having an l-th Peano derivative, g,, on I, g; is Darboux and if g; 2 0
on I then 9; = g“) on I . Suppose f is a function defined on R so that fk exists for
each x E R. Then there is a bilateral nonporous system of paths E = {E, : x 6 R}
satisfying the I. C. condition such that f], is the E-derivative of fk-1.

We will need some lemmas before we prove this theorem.

Lemma 2.1.10 Suppose that the assumptions of Theorem 2.1.9 hold. Then for every
6 > 0 and r] > 0 there is a 6 > 0 such that if] is a closed subinterval of (x — 6,x + 6)

with x not in I such that
Ifk-1(y) — fie-105)
y — x
for all y E I, then m(I) S 17d(x, I).

- fk($)| Z 6 (3)

Proof: Let 6 be chosen according to Theorem 2.1.5 applied with :7 replaced by
m = 17/(1 + 7]) and with j = k — 1. Let I be as above, and let g(y) = f(y) —

34

yk‘lé'fff—ffiz — (y - x)"£‘,§f-l. Then g has a (k — 1)—th Peano derivative and gk_1 (y) =
fk_1 (y) — fk-1(x) — (y — x) fk(x). So by assumptions gk_1 is a Darboux function. By
(3) ng-1(y)| Z cly — x| on I. Since x is not in I, ng_1(y)| > 0 on I and since gk_1
is a Darboux function, we have either gk—l > 0 on I or -gk-1 > 0 on I. Hence by
the assumptions, g]._1 is the (k — 1)-th ordinary derivative of g on I. Therefore f is
(k — 1) times ordinarily differentiable on I and by the uniqueness of Peano derivatives,
f(k'll = fk-1 on I. Now we can apply Theorem 2.1.5 with j = k — 1, which gives
that m(I) _<_ 171 - (m(I) + d(x, 1)). Hence m(I) S nd(x, I). C]

Next we will prove a lemma using ideas from the proof of 3.6.1 in [3].

Lemma 2.1.11 Under the assumptions of Theorem 2.1.9, for each point x E I there

is a path E3 leading to x and nonporous at x so that

lim fir-1U)" fk-IW) =

116131.31“: y — 13

fk($)-

 

Proof: For each 6 > 0 let 6(6) be as in Lemma 2.1.10 applied with n = 6/2 and let
{6;} be a sequence so that 0 < 6, _<_ 6(1/l) and 61+; < 61/2. Set

E:={x}u0{y>6z.2= f""(”)‘f""(””) -fk(x) <1/I}.

l=1 y_$

 

It is certainly true that

lim fk—1(3/)- fin-10")

véng-w y -— x

 

= fk($)-

(Although this assertion is true if x is not a point of accumulation of EL, in fact we
will prove that E; is nonporous from the right at x.)

Suppose E; is porous from the right at x. Then there must exist a number
1/2 < 0 < 1 and a sequence of numbers h; l 0 with [x + 0h;,x + h] n E; = 0

for every index I. Choose an integer lo larger that (1 — 0)"1 (i.e. so that if I _>_ lo,

35

then 1 — 1/1 > 0) and let jo be the first index for which hjo < 610. Fix I so that
61+, S hjo < 61, and note that I 2 I0.

Since hjo < 61 _<_ 6(1/1), by Lemma 2.1.10, there must be a point z with x + hj0(1—
’*-'<’2::*-1‘:1— fk(x)l < 1/1.

 

1/1) S 2 S x + hjo such that

 

We then have the inequalities,

x+61+2<x+%61+1<x+06¢+1 gx+0h,-o <:zc+(1—1/I)h,-o gzsx+h,-O.

 

From this, then, we see that (““(ztf'dfl — fk(x)[ < 1/1 and x + 61+; < 2, so

 

z E E;. But also x + 0th < z 5 x + hjo, so 2 E [x + 0hj0,x + hjo]. This contradicts
the fact that E; n [x + Ohj, x + h j] = 0 for all j. Similarly we define a path E;’ leading
to x that is nonporous from the left at x. The path EJr = E; U E;’ has the desired

properties. [I]

Now we are ready to prove Theorem 2.1.9.

Proof: For each x 6 R let E; be a path satisfying the conclusions of Lemma 2.1.11.
We will define the system of paths E = {Ex : x 6 R} as follows:

For x E R let E, = E; U P(f,1,6(x)) where 6(x) is such that x E P(f, 1,6(x)).
That E is nonporous (therefore bilateral) follows directly from Lemma 2.1.11. Also
Lemma 2.1.11 and Theorem 1.1.19, imply that fk_1 is E differentiable with
fk_1|;3(x) = fk(x) for every x E R. It remains to prove that E satisfies the intersection
condition I.C.. We will prove that for any two distinct points x and y, ExflEny, y] 9é
0 which is stronger than the LC. condition.

Let x and y be any two distinct points. If 6(x) S 6(y), then
P(f,1,6(y)) C P(f,1,5(x)) and hence y E E,. If 6(x) _>_ 6(y), then
P(f, l,6(x)) C P(f,1,6(y)) and hence x E Ey- Therefore Ex 0 E" f] [x,y] yé 0. Hence
E satisfies the LC. condition. This completes the proof of the theorem . E]

36

Now we will prove that we can drop the assumptions concerning the arbitrary
function g from Theorem 2.1.9. In order to do that we list some theorems from [3]

about path derivatives.

Theorem 2.1.12 Let E = (E, : x E R} be a system of paths that is bilateral and
satisfies the LC. condition. If f is an exact E-derivative and is Baire 1, then f has

the Darboux property.
Proof: This is Theorem 6.4 in [3]. D

Theorem 2.1.13 Let E = {E,, : x 6 R} be a system of paths that is bilateral and
satisfies the LC. condition. If _F_’E Z 0 on [a,b], then F is nondecreasing on the

interval [a, b].

Proof: See 4.7.1 in [3]. El

Theorem 2.1.14 Let E 2 {Ex : x 6 R} be a system of paths and suppose F is

monotonic. If E is nonporous at a point x, then

Ego) = E(x) and T’Ee) = 7’s).

Proof: See Theorem 4.4.3 in [3]. D

Theorem 2.1.15 Let f be a function defined on R and let lc E N. Suppose fk(x)
exists for each x E R. Then there is a bilateral nonporous system of paths E = {Ex :

x E R} satisfying the I. C. condition such that f], is the E-derivative of fk_1.

37

Proof: The proof is by induction on k. For k = 1 there is nothing to prove.
Let 1 S l S lc — 1, and let a function g defined on some closed interval I have a
l-th Peano derivative on I. Suppose the assertion of the theorem is true for every
1 S j S lc — 1, and every function h defined on some closed interval J, so that h,-
exists on J. (Note that we can restrict ourselves only to closed subintervals because
we can always extend h to R so that h j exists on R. For example if J = [a, b], then, we
can define h(y) = {=o(y — x)‘1‘§,fl for y E (—oo,a) and h(y) = {=o(y — x)‘£§,fl for
y E (b, 00).) By Theorem 1.1.7 g; is a Baire 1 function. By the induction hypothesis
and Theorem 2.1.12, g; is a Darboux function. Suppose that g; 2 O on I. Again by
the induction hypothesis but now using Theorem 2.1.13, g1_1 is nondecreasing on I.
By Theorem 2.1.14 g[_1 = g, on I. Also there is an a such that g1_1 — a Z 0 on I.
Let h(x) = g(x) — 0%. Then hz_1 = gz_1 — a and hence h1_1 Z 0 on I. Proceeding
as before h[_2 = h1-1 on I. This implies g{_2 = g;_1 on I. Continuing in this fashion

one can deduce that g“) exists on I. Now we can apply Theorem 2.1.9. Cl

Corollary 2.1.16 Let f be afunction defined on R such that fk exists for each x E R.

Then f]. is a Darboux function.
Proof: The assertion follows directly from Theorems 2.1.15, 2.1.12 and Theo-
rem 1.1.7. Cl

Definition 2.1.17 A perfect road of a function f at a point x is a perfect set P such
that

(I) x is a bilateral point of accumulation ofP

(2) f|P is continuous at x.

The assertion of the next corollary follows directly from the properties of Baire 1,

Darboux functions. (See [2])

38

Corollary 2.1.18 Let f be afunction defined on R such that fk exists for each x E R.
Then

(1) For each x, there exist sequences xn /' x and yn \, x such that fk(x) =
limit—too fl:($n) = limit-"00 fk(yn)-

(2) For each x

h(x) 6 [lizrgggf fk(z), limsup fk(2)] n [lizrggyf fk(2),1im 83p fk(2)]-

2-02- 2—03

(3) For each real number a, the sets {fk _<_ a} and {f1c Z a} have compact compo-

nents.

(4) The graph off). is connected.
(5) The function f], has a perfect road at each point.
(6) Each of sets {f1c < a} and {f}, > a} is bilaterally c-dense in itself. ( See [2].)

(7) Each of sets (f;‘ < a} and (f;‘ > a} is bilaterally dense in itself.

Definition 2.1.19 Let E = (E; : x E R} be a system of paths and F a function
on R. We say that F has the monotonicity property relative to E if for any interval
[a,b] the conditions FI’;(x) exists a.e. in [a,b] and Fl’;(x) Z a a.e. in [a,b] (resp.
Fax) 5 0) imply that the function F(x) — ax (resp. ax — F(x) ) is nondecreasing
on [a, b].

Theorem 2.1.20 Let E = {Ex : x E R} be a system ofpaths and let F be a function.
If E is bilateral and satisfies the intersection condition, and F is E -difl'erentiable, then

F has the monotonicity property relative to E.

39

Proof: See Theorem 6.6.1 in [3]. [3

Corollary 2.1.21 Let f be a function defined on R such that fk exists for each x E R.
Let [a,b] be an interval, and a be any constant. If f1, 2 a (or fk S a) on [a,b], then

a) fk_1(x) — ax (ax — fk_1(x) ) is nondecreasing and continuous on [a, b]
b) f“) exists and fl” = f,c on [a, b].
Proof: The assertion follows directly from Theorems 2.1.15, 2.1.20 and 2.1.14. El

Corollary 2.1.21 was first proved by Oliver in [10] and Corominas in [4]. See also
Verblunsky [15].

Definition 2.1.22 Let f be a function defined on R. If for any interval (a,b),
f‘1(a,b) # 0 implies m({x : f(x) 6 (a,b)}) > 0, then we say that f has the Denjoy
property.

Theorem 2.1.23 Let E 2 {Ex : x E R} be a system of paths and let F be an
E-difierentiable function that has the monotonicity property relative to E. If F g; is
Darboux Baire 1, then F}; has the Denjoy property.

Proof: This is Theorem 6.7 in [31°C]

Corollary 2.1.24 Let f be a function defined on R such that fk exists for each x E R.
Then f), has the Denjoy property.

Proof: The assertion follows directly from Theorems 2.1.15, 2.1.20, 2.1.23 and
Corollary 2.1.16. CI

Corollary 2.1.21 first was proved by Weil in [17].

40

Theorem 2.1.25 Let E = {E_., : x 6 R} be a nonporous system of paths satisfying
the intersection condition. Suppose that F is an E-difierentiable function with F 1’;
Baire I. If F}; attains the values M and -M on an interval Io, then there is a

subinterval I of Io on which F is differentiable and F’ attains both values M and
—M.

Proof: This is Theorem 8.1 in [3].

An immediate consequence of Theorems 2.1.25 and 2.1.15 is the following corollary

first proved by O’Malley and Weil in [12].

Corollary 2.1.26 Suppose fk(x) exists for all x in Io and let M 2 0. If fk attains
both M and —M on Io, then there is a subinterval I of Io on which fl. = f(k) and

f(k) attains both M and —M on I.

This corollary has some nice and immediate applications. The reader is referred

to [12] for the details and proofs.

CHAPTER III

In Chapter I we have shown that for any k-th Peano derivative, fk, defined on
R, there is a countable decomposition {Ha} of R into closed sets and a sequence of
differentiable functions {vn} so that for each n 6 N, v;ly,, = fk. In [10] O’Malley
showed that the same holds for approximate derivatives. Moreover he proved that
for any approximate derivative there is a decomposition of R into perfect sets with
the above property. In this chapter we will show that the same holds for Peano
derivatives. Also we will show that any k-th Peano derivative, fk, is a path derivative
of fk-1 with respect to a system of paths that is bilateral, nonporous and that satisfies
the internal intersection condition I.I.C.. This will enable us to give a positive answer
to the question posed by C. Weil, regarding the relationship between Peano and
selective derivatives, (See [19].) namely, the last result of this chapter is that f], is a

selective derivative of fk_1.

3.1 Relationship between f1. and fk_1

We will begin this section with a very well known lemma.

Lemma 3.1.1 Let n E N and let f and g be functions on R having n-th Peano
derivatives fn(x) and gn(x) at some point x. Then the function fg has n-th Peano
derivative at x and

fl

(fg)n(x) = 2 (:)f.(x)g.-.-(a=).

i=0

Proof: Let x 6 [a,b] be such that f,,(x) and gn(x) exist. Thus the following
formulas hold

41

42

f(x + h) = f(x) + hf1(-’€)+° +-,,-, n($) + h"€n(5r h)
9(32 + h) = 9(3) + h91($)+ + h—ignm + h" €n(x h)

where en(x, h) and En(x, h) tend toward 0 as h —9 0. Then

f(x+h)g(z+h)= 2 3,2; 'f‘ff)f}‘i(f))g+

h"e..(:c, h)g(x + h) + h"En(x, h)(f(x + h) — 5,,(x, h)) =
50% g ( )L-(z) )9..le +h"e:.(z,h)

where en(x, h) = 6,,(x, h)g(x + h) + €n(x, h)(f(x + h) — 6,,(x, h))

Since obviously linu._.o 6;,(x, h) = 0 we have

(mm) exists and (19m) = 2"; (;)f.(x)g.-.(x).g

i=0

Lemma 3.1.2 Let f and g be functions on R such that the n-th Peano derivative,
fn( x), and the n-th ordinary derivative, g(")(x), exist at some point x. Then

:(-1)’(2)(fg“’)n—j(x) = mum.

Proof: By Lemma 3.1.1

i<—1)j(:)(fgm).-.(x) =

:(-1)j(?) E: ("f-j)fi($)(9(j))(n_j-e)($) =
genie) Z (":’)f.-(x)g<"~"(x) =

n n—i

Z Z<—1)J‘ (3) ("?’)fa(w)g‘”“’(x) ——-

i=0j=0

43

n n—i

gig—1)"??? (i)fl(x)g‘""’(rv) =

n n-i

E (2‘) EH)" ("rhesus-”(s =

5:0 j=0

mean) + 33 (2)0 - 1)"“f.-($)g‘"“’(z) = semen

i=0

Lemma 3.1.3 Let H be a continuous function in an interval [a,b] containing 3;.
Suppose that H is n times Peano diflerentiable at each x 6 [a,b] and that H" is m
times Peano difl'erentiable at y. Then H is (n + m) times Peano differentiable at y,

and H(n+m)(y) = (Hn)m(y)-

This lemma was first proved by Corominas in [4]. We will use ideas of his proof
to prove this lemma, but before we give the proof we need some other properties of
Peano derivatives that are known. The following definitions and Lemma 3.1.5 are due
to Oliver. (See [10]). We will give a simpler proof of Lemma 3.1.5, than is given in

[10].

Definition 3.1.4 If f has an n-th Peano derivative at each point of an interval
[a, b], we say that f satisfies the mean value theorems M1,“, 1: = 0,1, . . . , n —l (or that
f 6 M5), iffor each x and x + h 6 [a,b], there is an x’ between x and x + h such

 

that:
ms + h) — fk(x) — h fig) _ ... _ z—fiili-ll)!.f.-1(x) = My). (1)
(n-k)!

When n = 1 and Ir 2 0, we have the ordinary mean value theorem for first
derivatives. The mean value theorem (Lemma 3.1.5 below) is that if fn exists on

some interval [a,b], then f 6 M3,”, 1: = 0,1, . . . ,n — 1.

44

The special case of M: when the left hand side of (1) equals 0, we refer to as
Rolle’s Theorem, Hi. In the usual manner, M: follows from Rf, by adding a suitable
polynomial to f. If f has an n-th Peano derivative on [a,b] and if y and y + h are

given in [a, b], for each x E [a, b] we set

my + h) — 2:: (gratis) . (x _ y)"

 

 

g(x) = f(x) _ hn'k/(n — k)! n!
It follows immediately that
hn—k-l
9:.(11 + h) - 9:.(31) - h9k+1(3/)-"' - (n _ k_1),gn_1(y)=0

i.e., that g satisfies the hypothesis of Hi; and that

fk(!/ + h) - 23:1: fifflyl
9’10”): fn(3’) " hn-k/(n __ k)! ’

 

Applying the conclusion of Rf, to g, i.e., replacing x by x’ and gn(x’) by O, the
conclusion of M]: follows for f.

It is also possible to deduce Rfi, k = 0,1,...,n — 2, from R”‘1 and M,’f_1, as

n

follows. We may write

fk($ + h) —' fk(x) - ' ‘ ° -' (—"_—::;k 22)Ifn-2(x)— (—_—::;:.11)gfn—l($)

hn-k/(n _ k)! =0

 

in the form
I (um-1 (r)----—(—;—W’:."."‘_".J.- (z)
k h:—h-I](n_k-1)§ 2 " fit-1(3) _ 0
h/(n — k) ‘
and replace the first ratio, using M: _1, to obtain fn-1(x ”)— fn_1(x)= O for some x”
between x and x + h. We use R3" to deduce from this last equation the existence of

15’ between x and x” for which fn(x’) = 0, the conclusion required by Rfi.

Lemma 3.1.5 Letn E N and let f be a function defined on some interval [a, b] so that
fn exists on [a,b]. Then fn satisfies the mean value theorems M,’f, k = 0,1,. . . ,n — 1,
on [a,b].

45

Proof: The proof is by induction on n. Since the assertion holds for n = 1, the
remarks before the lemma show that the induction will be completed by proving
f 6 R2“. So it is enough to prove that if fn-1(x + h) = f,,_1(x), then there is an 93’
between x and x + h for which fn(x’) = 0.

We may assume h > 0, because a proof for the case h < 0 is similar. If fn
is positive on [x,x + h], then, by Corollary 2.1.21, f("l exists and hence by Rolle’s
Theorem for derivatives there is an x’ between x and x + h so that f(“)(x’) = 0.
Similarly if f” is negative. If fn takes on both positive and negative values, then since

f" is Darboux, fn attains the value zero. The induction is complete. C]

Now we are ready to prove Lemma 3.1.3.

Proof: For each x E [a, b] let

T(1‘) ___ H(JS) _ $05—31 jHJ(y__)_ wi(_ y)n+i (Hn)i(y) (2)

i=0 i=1 (T! + 2)!
Then To) = My) = = any) = 0. Tum = H.(z)— 2.-..(x- vii—W"; "
and Tn(y) = (Tn)1(y) = - - ~ = (Tn)m(y) = 0. Since T 6 M2, for each x 6 [a,b] there

is a c, E [a, b] between x and y such that

 

Tm = (x ;!y)"r,.(c,). (3)
Since Tn(y) = (Tn)1(y) = - - . = (Tn)m(y) :2 0, we have
Tn(x) = (x — y)mem(y,x — y) where em(y,x — y) —> 0 as x -—) y. (4)

Combining (3) and (4) we get

T0”) = (x _ y)n+m%€m(yacx — 31):

Now ( 2) becomes

LI(—$) Z($_ y)j__ HJ(y) + Z($_ y)n+i (Hn)i(.y_____)+ +1:( __ y)n+mC ;+m(y,x _ y)

j=0 i=1 (n + 3)!

46

where (“mg/,3; __y) = fifigemwm, -— y) —§ 0 as x —+ y. This proves that Hn+m(y)

exists and equals (Hn)m(y) . EJ

Lemma 3.1.6 Let f be defined in an interval [a,b] containing 0. Suppose that the
lc-th Peano derivative of f at 0 exists, and that the l-th Peano derivative of f exists
on [a,b], where lc and l are positive integers with l 3 Ir — 1. Also suppose that
f(0) = f1(0) = = fk(0) = 0. Let g(y) = y‘(""). Then the function h defined by

h(y)=(:,)f(y —(;)/ mg was.
—1)’(i)/.y/.n"'2:f(t)g<'>(t)dt-~dx. fan/790,

and h(O) = 0 has an l-th Peano derivative on [a, b].

Moreover
w:{ i? ;;;:3

Proof: By assumption f (y) = ykek(0, y). Consequently all of the above integrals
are integrals of continuous functions. Hence h is well defined. Moreover for y 96 0,
y 6 [a,b] H(y) = f: (f2 3”" f(t)g(‘)(t) dt~ udxg i = 1,. . . ,l is i times ordinarily
differentiable and H(‘)(y) = f(y)g(‘)(y) for i = 1,...,l. By Lemma 3.1.1, fg“) is
l — i times Peano differentiable at y. Therefore by Lemma 3.1.3, H is l times Peano
differentiable at y and H¢(y) = (H(‘))1_;(y) = (f(y)g(’)(y))(;_,). Hence h is 1 times
Peano differentiable at y and

y)=§(-1)J (3190)) is)

and by Lemma 3.1.2, h1(y) = fz(y)g(y)-

47

It remains to prove that h)(O) exists and that h)(O) = 0. For y 96 0

hill = % {(é)y’ek(0.y)+(k-I)( (’) )/,,y""€k(0,t)dt+~-+

(k—I)(k—l+1)-(k—1)(,)/y/n-/%e,.(0t)dt .132}.

Hence lim——— h(y) =0. Therefore h(O) = h)(O) = = h)(O) = 0. E]

v-*0 y

Now suppose that f has an l-th Peano derivative in an interval [a, b] containing

x, and that fk(x) exists. Consider a function

T(y) = f(y) — f(x) - (y - x)f1(9=) - - (y - x) T

and its translate C(t) = T (x + t). Then G satisfies the hypothesis of Lemma 3.1.6
and by that lemma the function H defined by

H(y) = (3)G(y)g(y) - (1) joy G(i)g’(t) dt + . . . +
_1)l [) [01' /0x2 . - ,v/OW G(t)g(l)(t) dt . . . ([32 for y 7g 0

and H (0) = 0 has an l-th Peano derivative on x — [a,b]. Moreover by the same

9&9? if y if 0
= y -
lemma, H1(y) { 0 ify = 0 .
But Gz(t) = T)(t + x) = f:(t + x) — f)(x) — tf1+1(x)— —t"" ”£17,. Therefore we

have proved the following theorem.

Theorem 3.1.7 Suppose that a function f defined on an interval [a,b] containing a
point x has an l-th Peano derivative on [a, b] and a k-th Peano derivative at x, where

0 S l S h. Then the function F defined on [a,b] by

 

1()-";‘1'——31 (z) .
0 ify = x

is an l-th Peano derivative.

48

Corollary 3.1.8 Suppose that a function f defined on an interval [a,b] containing
a point x has a (k — 1)-th Peano derivative on [a, b] and k-th Peano derivative at x.
Then there exists a perfect set P C [a, b] of positive measure such that x is a bilateral

point of accumulation of P and

 

lim fk—1(y) ‘ flu—1(3)

146 P. y-w y —

= fk($)-

Proof: The function F from Theorem 3.1.7, applied with l = k — 1 is a (lc — 1)-th
Peano derivative and hence Baire 1, Darboux and has the Denjoy property. Therefore,
by Corollary 2.1.18 there is a perfect set H such that F is continuous at x with respect
to H. Since F has the Denjoy property there is a perfect set P of positive measure,
containing H, so that F is still continuous at x with respect to P. The set P satisfies

the assertion of the corollary. C]

3.2 Peano derivatives and Property Z

Property Z was introduced in [18] by Weil. He proved, that f), has the property Z at
every point of R. In [13] Marik gives a different proof of this fact. Moreover he proved
there that a k-th approximate Peano derivative has property Z. Also he generalized

this result to an assertion which when specialized to lc—th Peano derivatives is the

following theorem.

Theorem 3.2.1 Let j and k be integers, 1 S j S 1:. Let x E R and let f be a
function such that fk(x) exists. Define P(y) = f=o(y —x)’—‘-§,—‘52 for y E R. Let c > 0,
17 > 0. Then there is a 6 > 0 with the following property: If I is a subinterval of
(x — 6,x + 6) such that fj exists on I and that IfJ-(y) — P(j)(y)| _>_ ely — xlk‘j for all
y E I, then m(I) S nd(x,I).

49

For the special case of lc-th Peano derivatives, the proof of Mafik’s result is simpler
than for approximate lc-th Peano derivatives. Moreover the proof given here concludes
the case 1 S j S k — l as a consequence of the casej = lc. The casej = k, using

Theorem 3.1.7, is an immediate consequence of property Z for Peano derivatives.

Proof of Theorem 3.2.1: Casej = k. Let g(y) = f(y) — P(y), and let 6 > 0 be

such that
_ ml]:

(5)

 

my» < «liqr'y a

Let I be a subinterval of (x — 6, x + 6) such that

lgk(y)| Z 6 for y E 1- (6)

By the Darboux property, either gk(y) 2 e on I, or g), S e on I. By Corollary 2.1.21,

g“) exists on I, and hence by Lemma 2.1.7, there is a subinterval J of I such that

Igl 2 . - @915 (7)

Combining (5) and (7) give m(I) < fi—nly — x] for every y E J. Therefore m(I) S
T1? - (m(I) + d(x, 1)), hence m(I) S n - d(x, I).

Casej < k. By Theorem 3.1.7, the function hJ-(y) = W”- for y 9i x and
h,(x) = 0, is a j-th Peano derivative, and by what was just proved, for any 6 > 0 and
n > 0 there is a 6 such that whenever I is a subinterval of (x — 6, x + 6) and such that
|hj(y) — h,(x)| 2 e for y E I, then m(I) S 17 - d(x, I). This is exactly the claim of the

theorem for j < lc. E!

This theorem enables us to prove the following analogous of Theorem 2.1.15.

Theorem 3.2.2 Let k E N and let f be a function defined on R with f),(x) existing
for all x E R. Then for each integer 1 < r S k — 1 there is a bilateral nonporous
system of paths E = {E,, : x E R} satisfying the I. C. condition such that

— ’5‘"1.(!-_3li .3
h(x): lim fr”) 1:0 j! fr+J()

veExw-n' (y — x)""

 

50

The proof of this theorem is similar to the proof of Lemma 2.1.11 and Theo-
rem 2.1.9. Proof:

Let c > 0 and let 6(6) be the 6 from Theorem 3.2.1 applied with r] = 6/2. Let {6)}
be a sequence so that 0 < 6) S 6(1/l) and 6H1 < 61/2, and define the set E; by

00 Z _ 6:6'11flli r J
={x}UIL_J{z>x+61+2: fr( ) zf:-x)""' f+(x x)—fk(.’t) < l/l}.

 

It is certainly true that

k—r— 1 y-xJ It)
fk($) : “m fr(y) J: LTLfr+J()

yEE;.y—~z _(y — 2:)" "

 

Now we will prove that E; is nonporous on the right at x. Suppose not. Then
there must exist a number 1/2 < 0 < 1 and a sequence of numbers h; l 0 with
(x + 9h), x + h)) n E; = 0 for every index I. Choose an integer lo larger that (1 — 0)"l
(i.e. so that if I 2 lo, then 1— l/l > 0) and let jg be the first index for which h,0 < 610.
Fix l so that 614.1 S h,0 < 61, and note that I 2 lo.

Since h,o < 61 S 6(1/l), by Corollary 3.1.8 there must be a point z with x + (1 —
1/l)h,-0 S z S x + h,o such that

rz- k—r—1(Z_-j;_r_£rj$
m (2%)“ m )—f;.(x)

 

<1/I.

 

 

We then have the inequalities,

x+61+2<x+%61+1<x+06)+1Sx+0hjo <x+(l—1/l)h,-0 Ssz+h,-o.

k-r—l (2..in . 1‘
From this we see that f'(z)-E’::_x),_i‘ f'+’( ) — fk(x) < l/l and x + 6H2 S 2 so

 

z E E;. But also x + tho < 2 S x + hJ-o, so 2 E [x + 0h,0,x + hjo]. This contradicts
the fact that E; F] [x + 0h], x + hj] = 0 for all j. Similarly we define a path E;’ leading
to x that is nonporous from the left at x.

For each x E R we will define the system of paths E = {Er : x E R} as follows:

51

For x E R let E, = E; U E;’ U P(f,1,6(x)) where 6(x) is such that x E P(f,1,6(x)).
E is nonporous (therefore bilateral) follows from the fact that E; U E;’ is nonporous

at x, also Theorem 1.1.19 and what we proved in the first part assure us that

- ’-‘""Lflli .3
fk($)= lim f.(y) J=° J! fr+:()

yEEsm-n' (y — x)""'

 

for every x E R. It remains only to prove that E satisfies the intersection condition
I.C.. We will prove that for any two different points x and y, Ex 0 E, 0 [x,y] #9 0
which is stronger than the LC. condition.

Let x and y be any two different points. If 6(x) S 6(y), then P(f,1,6(y)) C
P(f,1,6(x)) and hence y E E1. If 6(x) Z 6(y), then P(f,1,6(x)) C P(f,1,6(y)) and
hence x E E,. Therefore E, D E, n [x,y] aé 9. Thus E satisfies the LC. condition.

This completes the proof of the theorem . D

3.3 Peano and selective derivatives

Recall the sets

13..

H(f,M,1)={$= 2

 

k—l
1fk(x) + Z(—l)"‘1‘j(k;1)j"ek(x,jt) S M for |t| < 1}

where M E N , from Definition 1.1.25. In Chapter I we showed that these sets are
closed, their union is R, and that with respect to these sets fk_1 differentiates to
fk(x) and Ifk(:l:)| S 2M for x E H(f,M,1). (See Theorem 1.1.27.) We have seen
several applications of this decomposition. Now we will prove that these sets can be
enlarged so that they are perfect, and that still with respect to these enlarged sets,
fk_1 differentiates to fk.

Let y E H(f, M, 1) be an isolated point of H(f, M, 1). Then there is al > 6(y) > 0
so that (y — 26(y),y + 26(y)) fl H(f, M, 1) = {y}. Let P, be a perfect set containing

y so that y is a bilateral point of accumulation of P, satisfying

lim fk-1(Z) '- fk-1(y)

ZEPyszfly Z - y

= fk(y)

 

52

and

fk-1(Z::;"-l(y) _ h(y) S 1 for every z E Py-

Corollary 3.1.8 assures the existence of P,. If P, H (y + "—11-,y + %) 9i 6 , for n 6

 

Z \ {—1,0}, then by the Baire category theorem there is a perfect set Qn(y) C
P, 0 (y + that] + fi), there is Mn(y) E N with Qn(y) C H(f,M,,(y), 1). Let
Qy = U Qn(y) n (y — 62(.11).:1 + 62(31)) U {y}. and let
n€l\{—l,0}

HM = H(f,M,1)U {Q,: y E H(f,M,1), y is isolated in H(f,M,1)}

Theorem 3.3.1 H M is a perfect set, and f)._1 is differentiable on H M relative to H M
with (fk_1|HM)’(x) = fk(x), for each x 6 HM.

Proof: By the construction of H M we see that no point is an isolated point. Note
that each Q, is perfect and that Q, 0 Q, = 0 if y,z E H ( f, M, 1) are two different
isolated points of H ( f, M, 1). Let {2,} be a sequence in H M such that limn_.°o 2,, = 2.
If 2,, E H(f,M,1) for infinitely many n, then 2 E H(f,M,1) since H(f,M,1) is
closed. Assume 2,, not in H ( f, M, 1) for each n E N. Then for each n 6 N there is
an isolated point yn 6 H ( f, M, 1) such that 2,, E Q,,,. If there are only finitely many
different y,,, then 2,, E Q, for infinitely many n. Since Q, is closed, 2 E Q, C H M.
Assume there are infinitely many different yn. Since Izn — ynl < 6(yn) < 1, and since
“an“, 2,, = 2, there is a subsequence {2",} of {2,} such that {ynj} converges. Let
y = limj.”o ynj. Then y E H(f,M,1) and it follows that z = lim,-_.co 2n]. = y. So
2 E H M» Therefore H M is closed.

Now if x E H M is an isolated point of H ( f, M, 1), then clearly f,’,_l at x relative
to HM, exists and is equal to fk(x). If x E Q, for some y E H(f, M, l) where y is an
isolated point of H(f, M, 1), then there is n E Z so that x 6 Qn(y) C H(f,M,,(y), 1)

53

and by the fact that there are two numbers a < b so that (a, b) 0 HM = Q,,(y), we

see that f,’,_1 at x relative to HM exists and is equal to fk(x).
Finally let x E H(f, M, 1), and x not an isolated point of H(f, M, 1).

Let e > 0 be given. Then there is e > r] > 0 so that

fk-Ify) — fie-1(3)
y — x

 

— fk(x) < 6

whenever y E H(f, M, l) and Iy — x] < 7}.

Let y be an isolated point of H(f, M, l) and let 2 E Q, with |z -—- x| < 17/2. Since
ly - 2| < 62(y) < 5(31) and Lu —e| > 25(31), we have 71/2 > le - 2| 2 le — yl - ly— 2| >
25(31) - 6(y) = 5(31). Hence ly — el S ly - 2| + lz - e| < 5(11) + n/2 < 77.

Thus

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

"‘4”: : ;"“(“’) — fk(x)l = (”“02 1"”) — h(x)) 3’ j “’+

(fern-1e) — h(y)) :1 + :: gm) — h(x)) s

fk-l(y3::k-l(x) — fk($)[l1— :1: +

"4‘": : f4“) - My) 41+ :1] use» + mm» s
my) .93) 62m

‘(H 6(y)) +1 6(y) + mums

26 + 6(y)(1+ 4M) 3 2.: +%(1+ 4M)

and since 6 was arbitrary we have that f,;_l at x relative to HM exists and equals
fk($). D

We end this chapter showing that a k-th Peano derivative is a path derivative of
the (k — 1)-th Peano derivative with a system of paths satisfying the I.I.C. condition.
As a corollary to this result we will obtain that a. k-th Peano derivative is a selective

derivative of the (k -— 1)-th Peano derivative.

54

To define the system (E, : x E R} of paths with respect to which a given k-th

Peano derivative, fk, is the path derivative of fk_1, we begin with some notation.

Notation For x,y 6 R let 6(x,y) = min{1, [lg—’11}. For x E R and M E N let
R, = U{P, n [y,y + 62(x,y)) : y E H(f, M, 1) and y is right isolated
from H(f,N,1) for N E N} and let
L, = U{P, n (y — 62(x,y),y] : y E H(f, M, l) and y is left isolated
from H(f, N, 1) for N e N}.

Definition 3.3.2 Let x E R. If there is an M, E N such that x is a bilateral point of
accumulation of H(f, M,,1), then let

E, = HMxUR,UL,.

If x is a right isolated point of H ( f, M, l) for every positive constant M but there
is an M, so that x is a left point of accumulation of H(f,M,, 1), or ifx is a left
isolated point of H ( f, M, 1) for every positive constant M but there is an M, so that
x is a right point of accumulation of H(f, M,, 1), let

E,=HM,UP,UR,UL,.

Finally if x is an isolated point of H (f, M, 1) for every positive constant M then
let M, = 1 and let

E,=HM,UP,UR,UL,.

Definition 3.3.3 Let E be the system of paths {E, : x E R}.

Lemma 3.3.4 Let k E N and let f be a function defined on R such that fk(x) exists
VII 6 R. Then E is bilateral and satisfies I.I. C. condition.

55

Proof: Clearly E is bilateral. We will prove a stronger condition than I.I.C. .
Namely we will prove that for any two points x and y E, n E, F] (x,y) .-,£ 0. Let
x < y be any two points. Suppose M, S M,,. If x is a right point of accumulation of
H(f,M,,1)C H(f,M,,l), then E,fl E,fl (x,y) 75 0.

If x is a right isolated point of H(f,M,, 1), then by choice of M,, x is a right
isolated point of H(f, M, l) for every M E N and x E H(f,M,, 1). Thus

0 e P..- n [m + 62m» n (x,y) c e. n E. n (x,y).

If M, > M, and if y is a left point of accumulation of H(f,M,, 1) C H(f,M,, 1)
then E, F) E, 0 (x, y) 75 0.

If y is a left isolated point of H ( f, M,, 1), then by an argument similar to the
above E, n E, n (x, y) 96 0. Therefore E satisfies the I.I.C. condition.

Cl

Theorem 3.3.5 Let k and f be as in Lemma 3.3.4. Then fk_1 is E differentiable
With f(k—l)E(-’”) = f),(x).

Proof: Let x E R, and e > 0 be given. Then there is an e > 17 > 0 such that

fk_1(y) - fie—10'?)
y — T

 

- fk(x) < 6

whenever |y — x] < r) and y E H(f,M,,l) or y E P,. Let 2 E E, be such that
[z - x| < 321. If 2 E P, for some y E H(f,M,,l) such that y is an isolated point
of H ( f, M, 1) from either left or right, and for every positive constant M, then 52‘ >
[z - x| _>_ Ix — y| — |y - 2| 2 26(x,y) — 6(x,y) = 6(x,y). Therefore Iy - x] S
Iy — z] + Ix — z] < 6(x,y) + 17/2 < 17. Hence

fk-1(y) - fie-1W)
y - x

 

- h(x) < e (8)

56

Thus
fk-1(Z) - fk-1($) _

Z—$

 

 

h(x) —f.<x))g:—:+

y—

 

 

: [(fk—1(y)- fir-1(3)

 

 

 

 

 

 

 

 

 

 

(fie-“Z: : 51.461) - h(y)) :1 + :::(f,(,) _ h(x)) S
hwy) : 5””) — h(x)] [1 _ :1 +
use) : [H(y) _ h(y) |:—:1]+ :::| (m(m + may)».

 

 

 

By (8), Theorem 1.1.27 and the relationship among x, y and z, the above inequality

is

 

 

6 Way) .6’(e,y) Way)
5 (1+a(..,y))+1 6(x,y) +6(,,,,4st

2e + 6(x,y)(1+ 4M,) 3 2e +§(1+ 4M,)

and since 6 was arbitrary we have that f(k_1)E(x) exists and equals fk(x). C]

Corollary 3.3.6 Let I: and f be as in Lemma 3.3.4. Then f), is a selective derivative
of fk-l .

Proof: Let a selection p(x, y) be defined as follows:
If x < y let p(x,y) = z, where z is any point in E, D E, 0 (x,y), if x = y, let
p(x, x) = x. Then for fixed point x0 we have

lim fk-1(P(1‘o,y)) - fk—1(1'0) = lim fie-1(2) — fk-1(10).

...... p(eom) - z. ...... z — ...,

 

 

Since 2 E E, we have that the above limit exists and equals fk(xo). D

 

CHAPTER IV

4.1 Decomposition of Generalized Peano derivatives

Definition 4.1.1 Let F be a continuous function defined on R, and let n E N. We
say that F is n-th generalized Peano diflerentiable at x E R, if there is a positive
integer q, and coefiicients F[,«](x), i = 1, . . . ,n such that for each h E R

q" (- 9+1)”-
F(— -(,q) )=£h"§—— 2,, hm- (Ff—ii! Hangman) (1)
J: j=0

where limh_.o eqq]n(x, h): 0.

Here Flo](x) = F(x) = F(°)(x) and Fl’j)(x) = f‘” F('j+1)(t) dt; i.e. F(‘jl is an
indefinite Riemann integral of the continuous function F (’j“) for j = 1, . . . , q. Note
that the definition of F[,-](x), i = 0, 1, . . . , n and of 631,,(x, h) don’t depend on which q-
fold indefinite Riemann integral, F ('0), of the continuous function F , is taken because
any two differ by a polynomial of a degree less than q. The above definition is the
same as the definition of (q + n)-th Peano derivative of a function F (‘4) at a point
x. Therefore by Lemma 3.1.3 the coefficients F],](x), i = 1, . . . ,n don’t depend on q,
either. The coefficient F[,,](x) is called the n-th generalized Peano derivative of the
function F at the point x. For the remainder of this chapter 12 will be a fixed positive

integer, and F will be a continuous function defined on R.

Definition 4.1.2 For q E N, let A, be the set of all x E R so that (1) holds, and for
e > 0, 6 > 0 let

Pa = P110515) = {e E A, |€q+n(e' h)I < 6. for I’ll < 5} (2)

57

58

Note that if x E A,, then x E A, for every p Z q. Also x E A, iff F(-q) has an
(q + n)-th Peano derivative at x with (F(‘9)),+,,(x) = F[,](x).

Lemma 4.1.3 For q S p, P,(e,6) C P,(e :1: !,6).

Proof: Let x E P,(e,6). Then x 6 A, and for t E R

(- Q+Jl
F(- (”(3 + t).— _ (’2th + Z tq+j F_lJ__l($)' + tq+n6 [,q.],n(x,t). (3)
j=0 1' j=0 (q + 1')

Integrating both sides of (3) from 0 to h we get

F(‘q’1)(x + h) — F(""1)(x) =

q-1 :H)——(—$) n
2),: th+1 +2}, hq+1+j___ +/h tq+n Min )dt.

Thus x 6 A,+1. By(] the remark after Definition 4.1.1, we have
hq+1+n 65:31,“ (x h) =/: tq+ne 5"],(x, t)d (4)

and since x E P,(e, 6) for 0 # |h| < 6 from (4) we have

|h|q+l+n

lhlq+l+nlclqfilil($a M] < jlh'tq+"e dt = e——— .
q n o q +1 + n

Hence |c[9331,(x, h)| < e/(q + n + 1) whenever Ihl < 6. Therefore

P,(e,6) c P,“ (adj: :)l)!’6) .

 

The general result follows by induction. [:1

Definition 4.1.4 For x E A, and fori = 1, . . . ,n, define egqlxx, h) by

F(-q 0+J')
F("’)(x+h)- 21.1—— flm +21: 11"” (Ffl—J—j) ), +h°+‘ elflxx, h). (5)
j=0 j=0

59

Note that eflfix, h) doesn’t depend on which q-fold indefinite Riemann integral, F (”1),
of F is taken.

The following formula follows directly from Definition 4.1.4.

Formula 4.1.5 Let x E A,. Then fori E N with 2 S i S n we have

ngl,_l($,t)= t(q quj——::}I+ “iii-T t)’

Recall Lemma 1.1.3, Definition of Riemann difference A?f(x) and Lemma 1.1.4
from Chapter 1.

Lemma 4.1.6 For m E N the following holds:

m . . 0 ifi=0,...,m—1
Z(—1)m-J(';:)j: = [ m! ifi = m

i=0 Ig-(m+1)! ifi=m+1

Definition 4.1.7 For any function f defined on R the Riemann difference Ag"f(x)
at a point x, of order m is defined by

AM =Z(-1)"’(’?fl)(e+fl>

The relationship between A? and A?“ is given by the following assertion.

Lemma 4.1.8 For any function f defined on R, for any m E N andt E R we have

A2"”‘f(e) = AI"f(=v + t) - A7705)-

60

Lemma 4.1.9 Let x E A,. Then for each i = 1, . . . ,n

A3+mF(—9)($) =

t°+mF1m1(x)+tq+m :3:3‘<—1)°+m-J' (IEM)j°+me£°lm(x.jt) ifm =z‘
,.+.~ ::r(-1)q+m-i(”jm)jv+‘e.~(x.jt) ifm > 2'.

Proof:
9+m .
A3+mF("')(x) = ;(_1)Q+m-J (qtm) F("’)(x + jt)
J:

= if (_1)q+m-I (07") (:0 0151:3121.

j=0 l=0

:0 W (F "1, ,——(%+ +(jt)q+‘e£°l.(x.jt))

1:0
115%- q+I)__(_,) 9+":

=1)”: ——,.—— zu- 1...-.(..-),.+

l=0 J=0
F (33)"+
q+l____ l’l 1)q+m—j q+m q+l
gt q+I)!,-§,((_1) (, )’ +
.Q+m .
tH-a Z(_1)q+m-J (9'37”)jq+i 41914:; jt)
i=0

which by Lemma 4. 1. 6, is equal to
tq+mF[m 1(2)) + tq+m zq+m(_1)q+m—J (9+m)jq+m€q+€l¢1lm($,jt) if m _____. Z
tq+i 29+m(_1)q+m-j (9+m)jq+i6q+l01i($, jt) if m > i. [1

Formula 4.1.10 Let x,x +t E A, and let i E N with 1 S i S n. Then

9+i+1
F[:1(9=+t)-Fm 2:) Z(- ,,.+.-+1_.(.+;;+1),.+. 511:: 36+

J=0

q-l-i q+i

P(_ 1) q+i—J (q+i)] q+i 651],” jt) _£(_ _1) q+i-J' (q-H), «1+5 file] +_(,, +t jt).

J=0

61
Proof: By Lemma 4.1.8,

A3+’+1F(‘°)(x) = A?+’F("’)(x + t) ._ A‘,’+’F(’q)(x). (6)

Applying Lemma 4.1.9, with m = i + 1 to the left hand side

and with m = i to the right hand side of (6) we get
q+i+1

tq+i z: (_1)q+i+1-.i (q+;+1)Jq+i [alga jt)-
=0
. n q+£ o . -
twig—1,1” + t) + W“ Z(_1)Q+t-J (9:1)jq+i£[Q]i(x + t,jt) _
i=0
s . q+£ . . . [ ]
tq+'F[,](x) _ tq+t Z(_1)q+t—J (Q:I)Jq+i€ 6:“(3 jt)

i=0

Dividing both sides by t?“ gives the above formula. C]
Theorem 4.1.11 For any interval [a, b], F[n] is bounded on P,(e, 6) f) [a, b].

Proof: Let [a, b] be an interval. Let x, y E P,(e,6)fi [a, b], so that for t = y —x we
have It] < 6 / (q+n+ 1), and let B: 2"?“ (q+g+1)jq+“. Then the right hand side of
Formula 4.1.10 applied with z = n, is bounded by 3315. It follows that F[,,] is bounded
on P ,(e,6)fl [a, b]. From Formula 4.1.5 (applied with i = n) it follows that for |h| < 6,
|eL°1,_,( . , h)| is bounded on P,(e,6) 0 [a,b]. Now from Formula 4.1.10 (applied with
i = n — 1) we see that F[,,_1] is bounded on P,(e,6) n [a,b], and again going back
to Formula 4.1.5 (applied with i = n — 1) we see that for |h| < 6, I491“, ( ,h)] is
bounded on P,(e, 6) 0 [a, b]. Continuing we can deduce that there is a constant C so
that |F[q(z)l S C for 1 S i S n, for x E P,(e,6) 0 [a,b].

Let x E P,, and let {xm} be a sequence in P,(e, 6) such that lim,,,_.,° xm = x.
Choose p 2 q such that x e A,. Let [a, b] be such that {xm} C P,(e, 6) n [a, b]. From
the first part of the proof we see that for l S i S 11, FM is bounded on P,(e, 6) n [a, b].

62

Therefore we can choose a subsequence {xmj} converging to x such that {F[,-](xm1)}
converges for each 1 S i S 11. Let these sequences converge to G;(x), i = 1,... ,n
respectively.
Let [h] < 6, and, as we may, suppose that |h+x —xm,I < 6 for everyj 6 N. Since
6 P,(e, 6), by Lemma 4. 1. 3 we have |ep+,,(xmj,h + x - xmj) )I < 612—3131. Thus we
may also suppose that the sequence ep+,,(xm1, h + x— me) converges. Denote its limit

by E(h). Now letting j -—-1 oo in the formula

F(-P)(m + h) : F(-P)(xm1) + (h + z _ 3m, )Ff-P+1)(xm1) + . . . +

(h+x—xm,)” (h+x—x,,,1)'°+""l
pl (p+n— l)!

+(h + 1' - $m1)p+n (

 

 

F[0](xm,) 'i" +

Fl"1($m1)
(p + n)'

F'[n—l]($m, )‘i'

+ e,+,,(xm1, h + x — me))

we get

1"“ h: p+1

_ hp
F( p’($+h 12? F’ ”m(x+';!$F[o]()+mG1($)+m+

(1):”: 1) G.-.( )+hp+" ((1)—70:“), +1300) (7)

+ E(h) 1s bounded, by the uniqueness of Peano derivatives from (7)

 

Gn x
p+n !

we have G,-(x) = Fm(x) for 1 S i S n — 1 and

Since

 

 

. + E(h). (8)

Since |E(h)|_ <e (—p———q + 2)! from (8) we have that

 

FM”)- Gn($) (____q+ n!)
(p+n)l =E'I (h) _£P+n(x’ h)|< — €(——P + n)! +|€ +CP+n($ h)i° (9)

The left hand side of (9) doesn’t depend on h so letting h —+ 0 in the right hand

 

 

side of (9) we get
F[n1(x) — Gn(x)

C(q + n)!
(p + n)!

_ (p+n)!'

 

 

 

(10)

63

Finally from the first part of the proof we know that there is a constant C, so that
|F[,,](y)| S C, for y E P,(e,6) 0 [a,b]. Since limjnoo F[,,](xm,) = G..(x), ICn(x)| S C.
Hence by (10) |F[,,](x)| S €(q+n)l+C. Note that the bound on F[n](1') doesn’t depend
on the choice of p. C]

If x1 and x are two different points in A,, then since F[,,](y) == (F(‘°)),+,,(y) for

y E A,, we have a formula for generalized Peano derivatives similar to the one in The-

n . Ia.
" "“1

orem 1.1.17. We will use this formula only for the case 3 = k — 1 in Theorem 1.1.17.

For the sake of completeness, we will state this formula for generalized Peano deriva-

._. 11... ~ .7—

tives as Theorem 4.1.12 below, and we will give a proof of this theorem not recalling

 

the corresponding result for Peano derivatives; i.e. Theorem 1.1.17. U

Theorem 4.1.12 Let x, x1 6 A, such that x 7b x1 andt aé 0. Then

Fln‘ll(x1)_ Fin-1K1.) _ 1;} 1(3) = t q + Tl —1

xl—x xl—x 2

11—1 n
9+2: (-—1)q+n—1-j(q+n—1) (5'31 - 37 + 3’)“ €[ql
J' tq+n—1(,,1 _ x) 0+"

 

 

F[n]($) 'l'

(x, x1 —x+jt)—
i=0

 

t q+n— ”n” n__- ,,_ .,, .
{ 2 F1n1(el)+ EH 1)"+ 1 ’("j 1)J°+€l°ln(e1,1t)}-

171-17

To prove this theorem we will need two technical lemmas.

Lemma 4.1.13 Let x1 6 A,. Then for any t

Ai+n-1F(-Q)($1) = tq+n_lF[n-1]($1)+

0+n-1

n +n_l n n——‘ n- n
t” q—E—Hn](x1)+tq+ Z (_1)q+ 1 J(Q+j1)jq+€ql<11n(xl,t)

i=0

64

Proof:
Q+n-1F(’IQ+) a)
A3+"’1F(“’)(x1) 2 Z (_ 1)q+n-1 -j (q+n—1=(Z(J) t)1______u( 1)+
i=0 l-O
" - 111(e $1) I]
Jt °+’—- +(J t6)"""‘ ° n(e .t
1:11;} ) (q + __l-_)l 611+ 1 )
11-1 q+n-1 n_ F(-q+l)(3
2( 1: <— ,,...- -,(.. )1) 1...." .) +
(:0 j=o °
n q+n-1 F ($1)
(_1)q+n—l—j q+n-1jq+l tq+l___ [’l +
g(_ z ( ) ( _, ,,.
°+"' - 11
,... z (_,,...-.-.(..,-1),...,.+n(,,,,,
i=0
By Lemma 4.1.6, the above is equal to
n__ n + n — 1
“H” ’11.-u(x1)4-t’* g-3;--Iiu(x1)4-
q+n-1 .
tq+n Z?) (_1)q+n—l-J(q+rJy—1)jq+nm[<11n(xh t) E]
J:
Lemma 4.1.14 Let x, x1 6 A,. Then for anyt
A:+"'1F(’°)(x1) _—_ tq+"‘1F[,,-1](x) +
— 1
tq+n-1(xl — :L‘)F[,,](:1:) +tq+nfl121—F[n]($) +
q+n—1
Z: (— 1) “n" ’J (“i"- )(x1 — x +jt)q+"e[,ql,,(x,x1 — x +jt).
J=0
Proof:
q+n-1
ATM—117(— q)(1:1)— _ Z (_1)q+n-l—J' (q+n-1)F(-9)(x + x1 _ x +jt)
J=0
9+n—l

= Z (_1)<1+n—1-J'(q+:—1),

i=0

 

65

(E(xl — (I? +Jt)’—(——_F H” )+ 2(31- 2 +Jt )’+’(q #1:?! +

1-0
(ml — . + me" was - x + jt))

q+n-1

= Z; (-1)”+"-1-’(q+?_1) (:(h — x + jt)—_—

lF(___—____ 9+f)($)+

f(x. -— x + jt)°+’——Fl”(x’ ) +

l=0 (q +1)!
q+n-1 .
Z: (_1)q+n—1-J («11-1-1’1-l)(:,.1 _ x +jt)q+" Chili-43,31 _ x +jt).
i=0
Since
q+n’1F-f 1+1)
2 (_ 1)q+n-l -.i (Ha-1) )C: ($1 —x+jt)’F (3) +
F, I!
" _ , q+l___ F1108) )
g(xi x +j ) (q +1) ,
q+n—1 q— —l l F(_,+1)
=2 (— 1)Q+n—1—J (0+n-1)Z(Z m()($l _ (Cy-«jt), n ($)+
1:0 :0 8:0 '
n q+l 17 (x)
9'“ $1 $)-Q+l -8 t [I]
gem- < >—..+m)
9‘1 I —s QFK—q+0(x)q+n-1 n— -' n- 's
=Zz(1_$)($l )—l )tT Z (_1)q+ 1 1(q+j 1)] +
l=08=0 ‘ 1:0
71 0+1 F( ) q+n— —1
9+1 2: _ III 0+l-8t8___ [ll( 1? 1) q+n—l -J (Hm—1 ,
g§(s)(l ) (q+l)l12(- (1)].
by Lemma 4.1.6 the above 18 equal to

 

n q-H )(m F100”) q+n— —1

Z Z (9:1 —x)"+"‘t‘(q+l)!1_ Z: (_1)q+n—l-j (q+n-1)ja.

l=n-l s=q+n—l
Applying Lemma 4.1.6 once more it is equal to

q+n—1
2

tq+””F[,,_1](x) + tq+"’1(x1 — x)F[,,](x) + t"""’ F[,,](x).

This completes the proof. [I

Proof of Theorem 4.1.12: The proof follows directly from Lemma 4.1.13 and

Lemma 4.1.14. 1::

 

66

Finally we are ready to prove the main result of this Chapter.

Theorem 4.1.15 Suppose for each a: E R F is n-th generalized Peano diflerentiable
at 1:. Then F[n—1] is diflerentiable on 75., = 75.,(£,6) relative to 75., with

Fin-III'FJI’) = Fm”)-

Proof: LetzEFq. Thereisap>qsothathAp. Let 1 >c’>0begiven.
There 13 0 < 17 < 6 such that |£p+n($, h)| < 6’ whenever |h| < 17. Let {mm} be a

sequence in Pq converging to a, so that [am — ml <-—"— p+ By Theorem 4.1.11, there is

n

a constant C so that |F[,,](:rm)| _<_ C, for every m E N. Let t = (mm — “6'71“". Then
ljtl < 5 and Iain. -$+jt| < 17, forj = O,1,...,q+n — 1. Therefore

Ielflna 2m — x +jt)| < e (11)
and by Lemma 4.1.3,
leyln(xm,jt)| < e for everyj = 0,1 .. . ,p+ n — 1 and for every m E N. (12)

Since 1:". E A,,, the formula from Theorem 4.1.12 gives

Fin-11(xm) - Fin—ll($)

 

 

 

 

 

x _ x — was) s gear—31.11“”), +
”fl (”‘2' )(1 72:1”: “”1 (x mm — x+jt)| +
j=o c p+n
65+» P—+’; 1'14me +2: (-1)’+"- _,( +"' )1?“ Human .
=0
By (12) and (11) together with Theorem 4.1.11 the above is
s 'avflLllnn] )1 + ,Zz; (”*2“)(1+je';+‘-)r+- +

£131:{P+_n___—IC+pEl(P+n—1)Jp+n6}.

i=0

 

67

Since 6’ was arbitrary we have

Flu—1]($m) - Flu—1](x)

— F[n](a:) —» 0 as mm 6 Pq, mm —> 2:.
zm—a:

 

Now for the general case let {2cm} be a sequence in P, such that 3:", —-> 2:. Let gm 6 R,
be such that Iym — zml _<_ Tiilxm -— xl and that

F(n—1](ym) — F[n_1](l'm)

ym"mm

 

— F[n](xm) S 1- (13)

 

 

By what was just proved, there is such a sequence ym. By Theorem 4.1.11, there is a

constant C such that |F[n1(:1:m)| S C for every m E N. This and (13) give

Flu—l](ym) — F[n—1]($m)

 

 

 

 

 

 

 

 

 

 

 

 

S. 0 +1. (14)
gm '- 3m
Now
Flu—1]($:) — flu—H(x) _ F[n](x) = Flu—11(xm) '— Flu-1](ym) mm - ym+
m "' 3m — 3117: 3m "" 33
Fin-11(ym) - F[n_11(x) ym - 1: mm - ym
{ ym _ 33 FM”) mm _ x - Fln]($) mm _ x
80 by (14)
Flu—11(3) fin-11”) ..W s
1 Fn— (ym) — Fri—1(a)) l 1
C 1 —— l ‘1 l l — Fn — —.
( + )m+ ym_z [1($)(1+m)+0m (15)

 

 

Finally since arm —-) 2:, ym -+ 1'. But gm 6 Pg, and hence by the first part

F[n-l](ym) "' Flu-110‘?)

ym-x

 

- F[n1(17) —* 0. (16)

Therefore by (15) and (16)

F[n-1]($m) ’ Flu-11(3)

zm‘x

 

-F[,,](:c) —+0as :cm 67’}, mm —* :c.

This completes the proof. D

 

68

Lemma 4.1.16 For each 6 > 0, U°° °° Pq(e,1/m) = R.

q=0 m=1

Proof: The assertion follows from Definition 4.1.2. D

Corollary 4.1.17 Suppose for each a: E R F is n-th generalized Peano differentiable

at 2:. Then Fla] is a composite derivative of F[n_1].

Corollary 4.1.18 Suppose for each a: 6 R F is n-th generalized Peano difi'erentiable

at 1:. Then Fin] is an approximate derivative of F[n_1] a.e..

Corollary 4.1.19 Suppose for each a: E R F is n-th generalized Peano differentiable
at 2:. Then Fin] 6 [A]’.

Corollary 4.1.20 Suppose for each a: E R F is n-th generalized Peano difl'erentiable

at at. Then F[n] is a Baire 1 function.

That FM is a Baire 1 function, was proved in [9]. The proof in that paper is not
as simple as the proof for Peano derivatives. Corollary 4.1.20 gives another proof of

this assertion.

4.2 Generalized Peano, path and selective derivatives

Next we will show that the following analogy of Theorem 2.1.9, holds for generalized

Peano derivatives.

Theorem 4.2.1 Let I E N with l S n — 1. Assume for each function g defined on
a closed interval I having an l-th generalized Peano derivative, gm, on I, 9U] is a
Darboux function and if 9U] 2 0 on I, then 9U] = g“) on I. Suppose F[n] exists on R.
Then there is a bilateral nonporous system of paths E = {E, : a: E R} satisfying the
LC. condition such that FM is the E-derivative of F[n__1].

 

69

We will need some lemmas before we prove this theorem.

Lemma 4.2.2 Under the assumptions of Theorem 4.2.1 for every 6 > 0 and n > 0

there is a 6 > 0 such that ifI is a closed subinterval of (x — 6,x + 6), a: is not in I
with
F[n—1](y) - Flu—1K9?)
y — a:
for all y e I, then m(I) S nd(:r,I).

 

— Flatt) 2 6 (17)

Proof: Let 6 be chosen according to Theorem 2.1.5, applied with 17 replaced by
171 = n/(1+ 1)) and withj = n — 1. Let I be as above, and let g(y) = F(y) —
yn‘lffi—Z’i—g? — (y — wring-:2. Then g has an (n — 1)-th generalized Peano derivative
and g[,,_1](y) = F[,,_1](y) — F[,,_1](:r) —- (y — x)F[n](z). So by assumptions g[,,_1] is
Darboux. By (17) |g[,,_1](y)| 2 cly — 2| on I. Since a: is not in I, |g[,,_1](y)| > 0 for
y 6 I and since g[n-l] is Darboux, we have either gln-” > 0 or —g[,,_1] > 0 on I. Hence
by the assumptions, g[,,_1] is an (n - 1)-th ordinary derivative of g on I. Therefore
F is (n — 1) times ordinarily differentiable on I and by the uniqueness of generalized

Peano derivatives, F ("‘1) = F[,,__1] on I. Now we can apply Theorem 2.1.5, with n = k

and j = n -— 1, which gives m(I) _<_ m . (m(I) + d(x, 1)) Hence m(I) S Ud($a1)- [j

The statement and the proof of a next lemma follow line by line the corresponding
Lemma 2.1.11 for Peano derivatives. So we will only state the lemma and omit the

proof.

Lemma 4.2.3 Under the assumptions of Theorem 4.2.1, for each point :1: E I there

is a path E, leading to a: and nonporous at a: so that

lim Flu-1101)" Fin—1K3) z Fin](x)-

yEExm-w y — a:

 

70
Lemma 4.2.4 Let m _<_ l be two positive integers and let 6 > 0. Then

1 1
Pm(€a —) C H(Ca—) '
m 1

Proof: By Lemma 4.1.3,

Pm(e, -1-) C P1(£, —1-), and by the Definition 4.1.2
m m

H(é’r—ii) C H(Cail- [:1

Now we are ready to prove Theorem 4.2.1.

Proof: For each a: E R let E; be a path satisfying the assertions of Lemma 4.1.9.
We will define the system of paths E = {E,, : a: E R} as follows:

For a: E R let E3 = E; U Pm(1,1/m) where m is a positive integer such that
a: E Pm(1,1/m). That E is nonporous (therefore bilateral) follows directly from
Lemma 4.2.3. Also Lemma 4.2.3 and Theorem 4.1.15, assure us that F[n—1] is E
differentiable with F[,,_1]|’E(z) = F[,,1(a:) for every 1: E R. It remains only to prove
that E satisfies the intersection condition I.C.. We will prove that for any two different
points a: and y, E3 0 EV n [:c, y] at 0 which is stronger than the LC. condition.

Let x E Pm(1,1/m) and y E P1(1,1/l) be any two distinct points. If m S I, then
by Lemma 4.2.4 Pm(1,1/m) C Pz(1,1/l) and hence a: E E”. Similarly if m 2 I, then
y E E3. Therefore E, 0 E3, 0 [x,y] 3‘5 0. Hence E satisfies the LC. condition. This

completes the proof of Theorem . E]

Next we indicate that we can drop the assumption concerning the arbitrary func-
tion from Theorem 4.2.1. The proof of that assertion follows line by line the proof
of Theorem 2.1.15, which is a corresponding result for Peano derivatives. So we will

only state the result.

 

 

71

Theorem 4.2.5 Let F be a continuous function defined on R so that Flu] exists on
R. Then there is a bilateral nonporous system of paths E = {Err : a: E R} satisfying
the I. C. condition such that Flu] is the E-derivative of Flu—1]-

Now using properties of path derivatives we get the following corollaries:

Corollary 4.2.6 Under the assumptions of Theorem 4.2.5 Fin] is Darboux.

Corollary 4.2.7 Let F be as in Theorem 4.2.5, let [a,b] be an interval and a E R.
IfFln] Z a (or F[n] _<_ a) then

a) F[n—1]($) - are (0:13 — F[,,-1](:r) ) is nondecreasing and continuous on [a, b]

b) F(") exists and F(") = F[n] on [a, b].

Corollary 4.2.8 Under the assumptions of Theorem 4.2.5 Flu] has the Denjoy prop-
erty.

Corollary 4.2.9 Suppose F[,,](a:) exists for all a: in Io and let M 2 0. If Flu] attains
both M and —M on Io, then there is a subinterval I of Io on which Flu] = F ("l and
F (“l attains both M and —M on I.

We end this chapter showing that every generalized Peano derivative Flu] is a
selective derivative of F[,,_1]. The idea is very similar to the one that we used for

Peano derivatives.

 

72

Definition 4.2.10 Let P1, be a set containing y so that y is a bilateral point of ac-

cumulation of P1, ,

lim Flu-11(2) "' Flu-11W)

zEPy.z-w z — y

 

and

Pin—11(2) "' Fin-1](y) __
Z — ll

 

Fln](y) S 1 for every 2 E P,.

Theorem 4.2.5 assures the existence of P,.
To define the system {Ex : :c E R} of paths with respect to which a given n-th
generalized Peano derivative, Fin], is the path derivative of F[,,_1], we begin with some

notation.

Notation For 2,3, e R let 6(x,y) = min{1, Ltgil}. For a: e R and M e N let
R, = U{Py F1 [y,y + 62(x,y)) : y E PM(1,1/M) and y is right isolated
from PN(1,1/N) for N 6 N} and let
L; = LJ{Py n (y — 62(z,y),y] : y E PM(1,1/M) and y is left isolated
from PN(1,1/N) for N E N}.

Definition 4.2.11 Let a: 6 R. If there is an M, E N such that x is a bilateral point
of accumulation ofPM,(1,1/M,), then let

E, = FM,(1,1/M,) u R. u L...

If a: is a right isolated point of 7540, 1 / M ) for every positive constant M but there
is an M,, so that a: is a left point of accumulation OfFM,(1,1/M,), or ifx is a left
isolated point ofPM(1, l/M) for every positive constant M but there is an M, so that
a: is a right point of accumulation ome,(1,1/Mx), let

E, = FM,(1,1/M,,.) u P, u R, U L...

 

73

Finally if: is an isolated point ofPM(1, l/M) for every positive constant M then

let MJr = 1 and let
E, =P’M,(1,1/M,)UP.UR.UL,.

Definition 4.2.12 Let E be the system ofpaths {Ex : a: 6 R}.

Lemma 4.2.13 Let n E N and let F be a function defined on R such that F[,,1(:r)
exists V1: 6 R. Then E is bilateral and satisfies 1.1. C. condition.

Proof: Clearly E is bilateral. We will prove a stronger condition than I.I.C. .
Namely we will prove that for any two points :1: and y E: n E, D (x,y) 75 0. Let
a: < y be any two points. Suppose Mr S M,,. If a: is a right point of accumulation of
FM,(1,1/M,) c Emu, 1/M,), then E, n Ey n (x,y) ;£ 0.

If a: is a right isolated point of PM,(1,1/M,,), then by choice of M,, :r is a right
isolated point of P540, l/M) for every M E N and a: E PM,(1,l/M,,). Thus

H P. n [x,x + 62m» n (x,y) c E. n E. n (x,y).

If M,,. > M,, and ify is a left point of accumulation ofPM,(1,1/My) C PMJI, l/MI)
then E: D Ey fl (x,y) at 9.

If y is a left isolated point of PM,(1,1/My), then by an argument similar to the
above E, n Ey n (9:, y) ;£ 0. Therefore E satisfies the I.I.C. condition. El

Theorem 4.2.14 Let F be a continuous function defined on R so that Flu] exists at
every point 1: 6 R. Then F[,,_1] is E difl'erentiable with F};_,]E(:r) = F[,,](:r).

Proof: Let :c E R, and e > 0 be given. Then there is an e > 17 > 0 such that

74

F[n—1](y) - F[n—l]($)
y — 2
whenever Iy — :c| < n where y E PM,(1,1/M,) or y 6 P1,. Let 2 6 E, be such

 

— PM“) < C (18)

that Iz — 2:] < 321. If 2 E Py for some y E PM,(1,l/M,) such that y is an isolated
point of PN(1, l/N) from either left or right, and for every positive constant N,
then 321 > '2 — 2:] 2 la: — yl — Iy — 2| 2 26(x,y) — 6(x,y) = 6(x,y). Therefore
|y — ml 5 |y — zI + In: — 2| < 6(x,y) + 17/2 < 7]. Hence by (18)

 

F“- -F,,_ a:
[ 11(y)_ [ 1]( )—F[,,](:c)
y z

 

 

< e. (19)
Thus
Fin—11(2) - F[n-1](-’B)

z—a:

 

— Flam

Z—IB

 

 

= “Flu-11h!) - Fin-11(3)

y — (I?
y _ 3 — Fin]($)) +

Z

 

 

 

 

 

 

 

 

 

Pin—11(2) - Fin—1K3!) Z — y " y
_ — <
( z _ y my) 2 _ x + Z _ m(m(m m(m» _
FINN”) " 1'1"“1”) — m(m) 1 — z - y +
y—x z—z
Fin-11(2) "' Fin—11(9) Z — y

 

 

 

 

- F[n](y)

 

 

 

 

+ z ' y I (|F[,,](:c)l + |F1n1(y)l)

z—y z—z z—x
By (19), Theorem 4.1.11 and the relationship between points x, y and 2 we get the
above inequality

Waxy)
506.31)

62W) + may)

3 c(1 + 50M!) 6(x,y)

 

)+1-

 

4M_<_

 

2: + 6(x,y)(1+ 4M) 3 2.: + £0 + 4M)

where M is a constant from Theorem 4.1.11. Since 6 was arbitrary we have that

FI’,,_1]E at :1: exists and equals to Fw(x) E]

 

75

Corollary 4.2.15 Let F be a continuous function defined on R so that Fin] exists at
every point x E R. Then Flu] is a selective derivative of Fin-1]-

Proof: Let a selection p(x, y) be defined as follows:
If x < 3] let p(x,y) = z, where z is any point in E, 0 Eu 0 (x,y), if x = y, let

p(x, x) = x. Then for fixed point x0 we have

lim F[::-1](P(1'o,y)) - F[n-1]($o) = lim Flu-11(2) — F[n-1]($O).

V“’° P(ico, y) - 30 ""° 2 - 550

Since 2 E E, we have that the above limit exists and equals F[,,](xo). CJ

BIBLIOGRAPHY

BIBLIOGRAPHY

1. S. Agronsky, R. Biskner, A. Bruckner and J. Marik, Representations of functions
by derivatives, Trans. Amer. Math. Soc. 263 (1981), 493-500.

2. A. M. Bruckner, Differentiation of real functions, Lecture Notes in Math., vol.
659, Springer, Berlin and New York, 1978.

3. A. M. Bruckner, R. J. O’Malley and B. S. Thompson, Path derivatives: A unified
view of certain generalized derivatives, Trans. Amer. Math. Soc. 283 (1984), 97-125.
4. M. E. Corominas, Contribution a la the’orie de la derivation d ’order supe’rieur,
Bulletin de la Société Mathématique de France 81 (1953), 176-222.

5. A. Denjoy, Sur l’integration des coefficients difl'erentiels d’order supe’rieur, Funda-
menta Mathematicae 25 (1935), 273-326.

6. M. Evans and C. E. Weil, Peano derivatives: A Survey, Real Analysis Exchange
7 (1981-1982), 5-23.

7. M. Laczkovich, 0n the absolute Peano derivatives, Ann. Univ. Sci. Budapest,
Ebtvbs Sect. Math. 21 (1978), 83-97.

8. C. M. Lee, 0n Absolute Peano Derivatives, Real Analysis Exchange 8 (1982-1983),
228-243.

9. C. M. Lee, 0n generalized Peano derivatives, Trans. Amer. Math. Soc. 275
(1983), 381-396.

10. H. Oliver, The exact Peano derivative, Trans. Amer. Math. Soc. 76 (1954),
444-456.

11. R. J. O’Malley, Decomposition of approximate derivatives, Proc. Amer. Math.
Soc. 69 (1978), 243-247.

12. R. J. O’Malley and C. E. Weil, The oscillatory behavior of certain derivatives,
Trans. Amer. Math. Soc. 234 (1977), 467-481.

13. J. Mafik, 0n generalized derivatives, Real Analysis Exchange 3 (1977-78), 87-92.
14. J. Marik, Derivatives and closed sets, Acta Math. Hung. 43 (1-2) (1984), 25-29.

76

 

77

15. S. Verblunsky, 0n the Peano derivatives, Proc. London Math. Soc. (3) 22

(1971), 313-324.
16. C. E. Weil, 0n properties of derivatives, Trans. Amer. Math. Soc. 114 (1965),

363-376.
17. C. E. Weil, 0n approximate and Peano derivatives, Proc. Amer. Math. Soc. 20

(1969), 487-490.
18. C. E. Weil, A property for certain derivatives, Indiana Univ. Math. J. 23

(1973/74), 527-536.

19. C. E. Weil, The Peano derivative: What ’3 known and what isn’t, Real Analysis
Exchange 9 No. 2 (1983-1984), 354-365.

20. A. Zygmund, Trigonometric series, 2’“ edition, Cambridge, (1959).

 

MICHIGAN sran UNIV. LIBRARIES
1M11H)1|)HIWMIllIHIIIHWIWHllllllllllll
31293007914751