WI MI [1 W \\ wk H RRR r l M W A DARBOUX PROPERTY FOR THE GRADIENT #4 4>o mm Thesis for the Degree of Ph. D. MicHlGAN STATE UNIVERSETY RAYMOND PETER GOEDERT 1972 This is to certify that the thesis entitled A DARBOUX PROPERTY FOR THE GRADIENT presented by Raymond Peter Goedert has been accepted towards fulfillment of the requirements for PhL D - degree in Mathematics QMUM Major professor Date July 28, 1972 0-7639 1:: mum av 110MB 8: SENS' 800K BINDERY INC. LIBRARY BINDEHS ABSTRACT A DARBOUX PROPERTY FOR THE GRADIENT By Raymond Peter Goedert A real-valued function f of a real variable is said to have the Darboux prOperty if f(I) is connected for every closed interval 1. A function is said to be Baire 1 if it is the pointwise limit of a sequence of continuous functions. In Chapter I of this paper we discuss the following generalization of the Darboux property. (It is due to A.M. Bruckner and J.B. Bruckner.) If X and Y are topological spaces and 8 is a basis for X, then a function f : X 4‘Y is said to be Darboux (m) if f(fi) is connected for all U 6 E. If X = Y = E1 (the real line) and 8 is the basis of Open intervals for 31' then this reduces to the usual notion of the Darboux property. C.J. Neugebauer characterized (in terms of inverse images of closed sets) the class of Baire 1 functions having the ordinary Darboux pr0perty. In Raymond Peter Goedert Chapter II we consider the family 8 consisting of Baire 1 functions mapping En (Euclidean n-space) into a fixed separable metric space Y. A theorem is proved establishing a condition which distinguishes those functions in 3 which are Darboux (Q), ‘where $ is any basis for En satisfying certain restrictions. The condition is stated in terms of inverse images of closed sets and is similar to Neugebauer's condition for Baire 1 functions having the ordinary Darboux prOperty. Examples are given showing the various hypotheses in the theorem cannot be omitted. An example is also given showing the theorem is no longer valid if the domain En is replaced by an infinite- dimensional normed linear space. A function f : E 4 E is said to be n l differentiable at X if f(y) - f(X) = (y-X)- grad f(x) + 0(\y-X\) as y 4 x. A function is said to be differentiable if it is differentiable at x for all x 6 En' In Chapter III we consider real-valued functions defined on En' each having a gradient everywhere in En' It is shown that the gradient of a differentiable function is Darboux (E), 'where 3 is any basis for En consisting of Raymond Peter Goedert connected sets satisfying a certain restriction. An example is given showing that this restriction cannot be omitted. An example is also presented showing that the gradient of a function which fails to be differentiable at even a single point need not be Darboux (B), ‘where m is any basis whatever for En' In addition. a Darboux prOperty for partial derivatives and an "intermediate- value property" for the norm of the gradient of a differentiable function are established. A DARBOUX PROPERTY FOR THE GRADIENT BY Raymond Peter Goedert A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1972 J ACKNOWLEDGEMENTS I wish to express my gratitude to Professor Clifford E. Weil for suggesting the problems studied in this thesis and for his kind guidance throughout its preparation. I also wish to thank Professor Jan Mafik for many most valuable suggestions. Finally I wish to thank my wife, Ann, for her patience and encouragement. ii TABLE OF CONTENTS Chapter I Introduction, Definitions, and Notation Chapter II ' A Generalized Darboux PrOperty . . . . Chapter III A Darboux Pr0perty for the Gradient . . Bibliography 29 4O CHAPTER I INTRODUCTION, DEFINITIONS, AND NOTATION Throughout this paper En will denote n- dimensional Euclidean space. The interior of a set S will be denoted by Int S, the boundary by Ed S, and the closure by §. A function f : E1 4 E1 is said to have the Darboux preperty if. given any a and b, a < b, and any c between f(a) and f(b), there is some t 6 (a,b) such that f(t) c. (This defining pr0perty is usually termed the intermediate value property. A detailed exposition of functions with this prOperty can be found in the survey article [3].) This definition gave rise to the following generalization due to Misik [5]. Definition 1.1. Let X be a tOpological space, and let 8 'be a basis for X. A function f : X 4 E1 is said to have the strong Darboux pr0perty relative to B if, given any U E E and any real number c for which there exist x1 and x2 in 5 with f(xl) < c < f(xz), there is some t 6 U such that f(t) = c. Notice that, if X = El and T is the basis for El consisting of all open intervals, then this reduces to the ordinary Darboux property. Now the main purpose of this paper is to establish a Darboux property of some sort for the gradient. Thus Misfk's generalization is not the apprOpriate one, for it requires the functions under consideration to be real-valued which the gradient is not. The ordinary Darboux pr0perty for a real- valued function of a real variable can also be defined in the following fashion. (This definition is equivalent to the one given above.) A function f : E1 4 E1 is said to have the Darboux property if f(I) is connected for each closed interval I. This definition gave rise to the following generalization due to Bruckner and Bruckner |2]. Definition 1.2. Let X and Y be t0pological spaces, and let S 'be a basis for X. A function f : X 4'Y is said to have the Darboux property relative to B (or, more briefly, is said to be Darboux (3)) if f(fi) is connected for all U E 8. It is this generalization of the Darboux prOperty which will be used in this paper. Notice that, if X = Y = El and 3 is the basis for El consisting Of all Open intervals, then this reduces to the ordinary Darboux prOperty. Suppose now that 8 is a basis for a t0pologica1 space X. Then, if a function f : X 4 E has Misik's 1 strong Darboux property relative to 3, it is trivial to show that f is also Darboux (3). However, as the following example shows, the converse is not true. Example 1.1. Let 8 be the basis for E2 consisting of the Open rectangle R ‘with corners at (0,1), (-1,l), (-1,-1), and (0,-1) and all Open balls not tangent to the y-axis. Define f : E2 4 El as follows. ' 1 5111 (m , X < -l O , -1 3.x < O f(x,y) = z 0 , x = O and \y‘ > 1 y , x = 0' and ‘y‘ g 1 . 1 L31n (i) , x > O It will be shown in Example 2.11 that f is Darboux (B) . But f does not have the strong Darboux prOperty relative to e, for f(R) = [-1,1] while f(R) = {o}. There are certain terms and notational conventions which will be used throughout this paper. They are presented at this point. Definition 1.3. Let (X,d) be a metric space. Then an Open ball is denoted and defined as follows: B(a,r) = {x E X : d(x,a) < r}. Definition 1.4. A topological space is said to be separable if it contains a countable dense subset. Definition 1.5. A function is said to be Baire 1 if it is the pointwise limit Of a sequence of continuous functions. Definition 1.6. A function f : Bn 4 E1 is said to be differentiable at x if f(y) - f(X) = (y-X)- grad f(X) + 0(ly-X\) as y 4 x, where - denotes the usual scalar product in En and grad f(x) is the vector whose ith coordinate is the partial derivative of f with respect to the ith variable evaluated at x. The function f is said to be differentiable if it is differentiable at x for each x 6 En' Neugebauer [6] characterized (in terms of inverses images Of closed sets) the class of Baire 1 functions having the ordinary Darboux property. (Also see Weil [7].) In Chapter II we consider the family 3 consisting of Baire 1 functions mapping En into a fixed separable metric space Y. A theorem is proved establishing a condition which distinguishes those functions in g which are Darboux (m), where E is any basis for En satisfying certain restrictions. The condition is stated in terms Of inverse images Of closed sets and is similar to Neugebauer's condition for Baire 1 functions having the ordinary Darboux property. Several examples are given demonstrating that the various hypotheses of the theorem cannot be omitted. In particular, a rather lengthy example is presented showing that the domain En cannot be replaced by a general normed linear space (in fact, not even by a separable Banach space) if the theorem is to remain valid. There is a theorem due to Bruckner and Bruckner [2] establishing another condition which distinguishes those functions in 8 ‘which are Darboux (m), 'where E is any basis for En satisfying certain restrictions. The condition is stated in terms of inverse images of open sets and is similar to a condition due to Zahorski [8] for Baire 1 functions having the ordinary Darboux property. Bruckner and Bruckner's result follows quite easily from the theorem mentioned in the last paragraph and is presented as a simple consequence thereof. In Chapter III the results Of Chapter II are applied to the gradient. (When we consider grad f here, we tacitly assume f is such that grad f exists everywhere in En') An example is presented showing that the gradient of a function which fails to be differentiable at even a single point need not be Darboux (S), where E is any basis whatever for En' Using a theorem from Chapter II, it is shown that the gradient of a differentiable function is Darboux (3) provided each element in 8 is connected and none comes to a cusp on its boundary. The major result in Chapter III has some rather interesting corollaries. In particular, a Darboux property is established for partial derivatives. In addition, it is shown that the norm Of the gradient of a differentiable function has Misik's strong Darboux property provided certain restrictions are placed on the basis for En under consideration. CHAPTER II A GENERALIZED DARBOUX PROPERTY In [7] Weil defines a ball closed G6 set as follows. A Go set H C En is called a ball closed 66 set if, whenever B(x,r) CZH, one has Bd B(x,r) crH. Neugebauer proved in [6] (without using Weil's terminology) that a real-valued function f Of a real variable is Baire l and has the ordinary Darboux property if and only if the sets [x : f(x) 2Da} and [x : f(x) g_a] are ball closed G6 sets for each real number a. In this chapter we shall consider the family 8 consisting of all Baire 1 functions mapping En into a fixed separable metric space Y. As remarked earlier, the main Objective of Chapter II is to establish a condition distinguishing those functions in 3 which are Darboux (E), 'where S is any basis for ED satisfying certain restrictions which will be introduced later. The condition will be stated in terms Of inverse images Of closed sets and will be similar to Neugebauer's condition for Baire 1 functions having the ordinary Darboux prOperty. TO establish this condition it is first necessary to isolate the "ball closed" part of the ball closed G concept. 6 Definition 2.1. Let X be a topological space, and let 8 ‘be a basis for X. A set S CLX is said to be D closed if, for each U 6 B, Bd U c: 8 whenever U c S. Example 2.1. Clearly any closed set in En is 8 closed no matter what the basis B is. However, a set can be S closed without being closed. Let N 'be the Open—ball basis for E2. Let S: [(x,y) : ogx_<_1, O_<_ygl, (x,y) 7! (1.1)}. S is not closed, but S is B closed. Example 2.2. Even an Open set in En can be closed. Let B be the basis for E1 consisting Of all Open intervals except those having 0 as a right endpoint. Then (-w,O) is both Open and B closed. Example 2.3. A set which has no interior is vacuously B closed no matter what a is. For example, in E1 both Q (the set Of rationals) and El o'Q are 8 closed for every basis 3. Most of the results in this chapter require that certain restrictions be placed on the bases under consideration. The three restrictions that will be used are presented here with some discussion. Definition 2.2. Let X be a metric space, and let 8 be a basis for X. Then 8 is said to satisfy (1) if, given any Open ball B and any x 6 Bd B, there is some V E 8 such that V CIB and x 6 Bd V. Definition 2.3. Let X be a metric space, and let T be a basis for X. Then 8 is said to satisfy (2), if, given any U E E and any x e 6, there is some V 6 8 such that V ~I[x] CIU and x 6 9. Definition 2.4. Let X be a metric space, and let E be a basis for X. Then E is said to satisfy (3) if, given any U e B and any x e U, there is some V e B such that V C‘U and x 6 Bd V. Condition (1) says that, given any ball and any point on its boundary, you can touch that point with a basis element lying inside the ball. Condition (2) says that, given any basis element U and any point x in its closure, you can touch x 'with a basis lO element whose closure (except possibly for x) lies inside U. Condition (3) says that, given any basis element U and any point x in U, you can find another basis element contained in U having x on its boundary. Conditions (1) and (2) are independent as Examples 2.4 and 2.5 show. Conditions (2) and (3) are independent as Examples 2.5 and 2.6 show. Example 2.7 shows that condition (3) does not imply condition (1). In general, condition (1) does not imply condition (3) either. For example, any basis for a discrete space satisfies (1) but not (3). However, if B is a basis for En which satisfies (1), a must also satisfy (3). Example 2.4. Let T be the basis for E2 consisting Of all Open balls and the Open rectangle R with corners at (0,1), (-1,l), (—1,-l), and (O,-1). 8 satisfies (1), for clearly, given any ball and any point on its boundary, you can touch that point with another ball (which will be a basis element) lying inside the given ball. But B does not satisfy (2), for you cannot touch any corner x of the rectangle with a basis element whose closure (except for x) lies inside the rectangle. 11 Example 2.5. Let 8 be the basis for E consisting of all Open intervals not having 0 l as an endpoint. B clearly satisfies (2). However B does not satisfy (3). For example, (-1,1) is a basis element and O is in its interior. But there is no basis element lying in (-l,1) having 0 as a boundary point. Notice that B does not satisfy (1) either. Example 2.6. Let E be the basis for E2 consisting of the Open rectangle R With corners at (0,1), (-l,l), (-l,-l), and (O,-1) and all Open balls not tangent to the y-axis. E satisfies (3), for clearly, given any basis element and any point x inside it, there is an open ball which is not tangent to the y-axis (and, hence, is a basis element) lying inside the given basis element and having x as a boundary point. However 8 does not satisfy (2), for you cannot touch any corner x of the rectangle with a basis element whose closure (except for x) lies inside the rectangle. Example 2.7. Let S ‘be the basis for E1 consisting of all Open intervals except those having 0 as a right endpoint. Clearly E satisfies (3) but not (1) . 12 Remark. Most Of the results in this chapter require that at least one of the three restrictions mentioned above be placed on the basis E under consideration. It should be noted that, if B is the basis for ED consisting Of all Open balls, 3 automatically satisfies (1), (2), and (3). However, as Bruckner and Bruckner mention in [2], this Offers no help in overcoming the difficulties caused by other bases. The following theorem is well known. It was originally proved (in an equivalent form) by R. Baire. (See, for example, [4, p.326].) It is presented here without proof. Theorem 2.1. Let X be a complete metric space and Y a separable metric space. Let f : X 4‘Y be Baire 1. Then, for any non-void G set S CJX, 6 f[S (f restricted to S) has a point of continuity. The following theorem is the major result in this chapter. Theorem 2.2. Let Y be a separable metric space. Let f : En 4‘Y be Baire 1. Let B 'be a basis Of connected sets for ED satisfying (1). 13 1 Then, if f- (K) is E closed for all closed K C'Y, f is Darboux (s). Proof. Let f-1 (K) be E closed for all closed K c'Y. Suppose however that f is not Darboux (E). Then there is some U e E such that f(fi) is not connected. Hence there exist sets A* and B*, closed in Y, such that ._ - * .. * f(U) = (f(U) DA) U (f(U) n B) is a decomposition of f(U) into two disjoint, non- void, relatively closed sets. _ * Let A=[x€U:f(x) EA}. Let _ * _ B=[x€U: f(x) EB]. Notice that U=AUB. Let P = U n Bd A. Notice that P = U 0 Bd B also since A and B are disjoint, U C’A U B, and U is Open. It is claimed that P # ¢. for suppose P is void. Then each x e U is in either Int A or Int B. Hence, since U is connected, either U C.A or U CtB. If U CIA, Bd U CIA since f_1(A*) is E closed. Hence B = ¢ ‘which is a contradiction. If U c.B, Bd U CJB since f-1(B*) is E closed. Hence A = ¢ 'which is a contradiction. 14 Notice that P is clearly a G6 set. It is claimed that A 0 P is dense in P, for suppose not. Then there exists some b0 6 P and some Open ball N = B(bo,r) C‘U such that N D (A n P) = ¢. That is, there are no points of A H Bd A in N. In particular, bo (’A n Bd A; so b0 6 B. Since bO E B 0 Bd A, there exists some a0 6 A 0 B(bo'§)' Let d = dist(aO,E). Then r Oc is in B, an identical argument again leads to a contradiction.) Hence f must have been Darboux (E). The hypothesis in Theorem 2.2 that f be Baire 1 cannot be omitted as the following example shows. Example 2.8. Consider E1 with basis E consisting Of all Open intervals. Let f be the characteristic function Of the rationals Q. (Then f is Baire 2 but not Baire l.) The basis certainly satisfies (1). The inverse image Of any closed set (in fact, of any set) is either Q, El ~'Q, ¢, or E1 and is therefore E closed. However f is clearly not Darboux (E). The hypothesis in Theorem 2.2 that the basis E for En satisfies (1) cannot be omitted (or even replaced by the weaker condition (3)) as the following example shows. 16 Example 2.9. Consider E1 with basis E consisting of all Open intervals not having 0 as a right endpoint. As was stated in Example 2.7, E satisfies (3) but not (1). Let f be the characteristic function of [0,m). Clearly f is Baire l. The inverse image of any set is either (—w,0), [0,m), ¢, or E1. Hence the inverse image Of any closed set is E closed. However, f is clearly not Darboux (E). One might ask whether in Theorem 2.2 one could replace En Tby a normed linear space of some sort. It happens that, as the following example shows, replacing En even with a separable Banach space invalidates the theorem. Example 2.10. Consider co, the space of all sequences Of real numbers converging to 0. The norm on C0 is given by “x” = sip [xk\, where x = {xk} 6 co. The space cO is a metric space with metric generated by the norm. Let E 'be the Open-ball basis for co. Clearly E satisfies (1). It is well-known that CO is a separable Banach space. 17 Let ei be the sequence with a 1 in the ith place and 0's elsewhere. Let S be the collection of all finite rational combinations of the ei's. It is easy to show that S is countable. To show that S is dense, let x = [xkl 6 c0 and e > 0 be given. Choose K sufficiently large that ‘xkl < e for all k.2 K. Choose rationals r1,...,rK_1 such that ‘Xk-rk‘ < e for k = 1,...,K-l. Let s = (r1....,rK_l,0,0,...). Note that s E S and that Hx-s” < 6. Hence S is indeed dense in co. Now let -k H = {x = [xk} E c : Z) x 2 = 0]. It is claimed that H is closed. Let c > 0 and y = {yk} E H be given. Then there is some x = [xk] E H a) such that Hy-xH < 6. Since 23 ka-k = 0, one has k—l that m —k m -k m -k \ Z} y 2 = Zi(y -x )2 ;_ 23 y -x \2 k=1 k ‘ ‘k=1 k k ‘ k=1‘ k k _<_ Hy-xH 2: 2‘k < e. k=l But 6 was arbitrary. Hence 2) y Z-k = 0. Hence k=1 k y E H. Hence H is indeed closed. Observe that, if a g H, then dist (a,H) > 0. This follows immediately from the fact that cO is a 18 metric space and H is closed. Observe also that, if z = [zkl 6 c0 and z # 0, then Q 1 E zk2“k\ < Hz“. To see this, note that [zk] < Hz” for all k sufficiently large since 2k 4 0 as k 4 m. Hence one has that 1 E z 2"k\ g élxzklz'k < Hzné’:l 2'k = 1le- k=1 k It is claimed that dist(a,H) = I Z) ak2—k[, k=1 where a = {ak}. Let x = [xk} E H. Then, since Z} ka-k = 0, one has that k=1 ” -k m —k m —k [ZaZ =[Z(a-x)2 gZa-xp k=1 k ‘ k=1 k k ‘ k=1‘ k k m -k s. Ha-xllZ 2 = Ila-x. k=1 Hence “ -k . . \ Z3 ak2 ] g inf Ha-XH = dist(a,H). k=1 xgH TO show the reverse inequality, let el = lei} be as defined above. Let 2m(2m_l)—l Z} akZ-k. k=1 Am Let m X m . m _ _ 1 i=1 m Note that xm E co. It is claimed that x E H. a) m 03 Z XEZ-k — Z (ak-Am)2 k + Z ak2 k k=1 =1 k=m+l m - m - k=1 k=1 2'm(2m-1)A - A 2'm(2m-1) = o. m m Hence xm e H. One has that .§? eiu = \Amx u,§> ein i=1 i=1 —1 -k [Am] = 2m(2m-1) (RE: k2 \. na-xmn = uAm Therefore dist(a,H) g Ha-me = 2‘“(2“‘-1)'1\ 2: a 2 k Letting m 4 m, one gets dist(a,H) 3" Z) ak2 =1 If a E'H, there is IND x E H such that dist(a,x) = dist(a,H), for suppose there were such an x. Then Ha-XH dist(a,x) = dist(a,H) -k k E: 2 = E: ( - )2" ‘k=l ak ‘ ‘k=1 ak Xk ‘ which is a contradiction since Ha-XH # 0. Now let S and e1 be as defined above. S ~lH is countably infinite since S is countably infinite and e1 e S ~'H for i = l,2,--o. Hence 20 . ~ 1 2 . n one can write S ~'H = [S ,s ,...). Since 3 A H, . n n . n dist(s ,H) > 0. Let Bn = B(s ,dist(s ,H)). Note that fin c c «:H, for, if not, there would be some 0 x E H with dist(sn,x) = dist(sn,H) which is impossible. Define a continuous function fn on a closed subset of co as follows. n - 1, x E L) E f (x) = n OI X E H Next use the Tietze Extension Theorem to extend fn continuously to all Of co. Denote the extended function by fn' DO this for each n = 1,2,'°-. Note that fn(x) = 0 for each x 6 H; so f(x) = 1im fn(x) exists and equals 0 on H. It is n4m ~ claimed that f(x) = 1im fn(x) exists and equals 1 ndm on C0 ~'H. SO suppose that a K H. Then dist(a,H) > 0. Since S is dense in c0 and B(a,% dist(a,H)) n H = ¢I there is some n such that sn 6 (S ~'H) fl B(a,% dist(a,H)). But then dist(sn,H) > dist(a,H). Hence dist(a,sn) < dist(a,H) < dist(sn,H). NH‘ NH‘ Therefore a E Bn' and, hence, f(a) = l. 21 It is claimed that f-1(K) is E closed for all closed K c E . Clearly it suffices to show that 1 f-1(l) and f-1(0) are each E closed. First consider f_l(0) = H. Since H is closed, f-l(0) is clearly E closed. Next consider f-1(l) = CO “’H. 1 Let U = B(z,r) c f- (1). U is in 9:. Let x e Bd U. Then f(x) = l, for, if not, one has that x E H and dist(z,H) = dist(z,x) which is impossible since 2 f H. Hence 6 C f—1(l). Hence f-l(l) is E closed. Finally, it is clear that f is not Darboux (m). The unit ball, for example, is a basis element. But f maps its closure onto [0,1], which is not connected. If the restriction on the basis is changed, one can prove a converse to Theorem 2.2 even without the assumption that f is Baire 1. Theorem 2.3. Let Y be a metric space. Let E be a basis for En satisfying (2). Then, if f : En 4y is Darboux (s). f-1(K) is :3 closed for all closed K c‘Y. Proof. Let K CiY be closed. Let U be any basis element contained in f-1(K). It must be shown that Bd U c: f'1 1< (K). Let x 6 Bd U, but suppose that x £ f_ K). Since E satisfies (2), there is 22 some V E E such that x e V and V ~ {x} C‘U c f.-1 (K). (The set V ~'[x} cannot be void, for V is an Open subset of En') Hence f(V ~'[x]) c K, and f(x) 5 K. But then f(V) = f(V ~'[x}) U {f(x)} is a decomposition Of f(V) into two disjoint, non—void, relatively closed sets. Hence f(V) is not connected which is a contradiction since V E E and f is Darboux (E). The hypothesis in Theorem 2.3 that E satisfies (2) cannot be omitted even if one assumes that f is Baire 1 as the following example shows. Example 2.11. Let En = E and let 2' Y = E1. Let E be the basis for E consisting Of 2 the Open rectangle R with corners at (0,1), (-l,1), (-l,-1), and (0,-l) and all Open balls not tangent to the y-axis. As was mentioned in Example 2.6, E does not satisfy (2). Define f : E2 4 E1 as follows. r . 1 Sin (X+l , x g —1 O o "ISXKO f(x,y) = fl 0 , x = 0 and [y] > 1 y , x = 0 and [y] g_l . 1 KSin (i) , x > 0 It is easy to show that f is Baire 1. The function f is also Darboux (E). Any ball whose closure lies to the left of x = -l or to the right of':x = 0 automatically 23 has its closure mapped onto a connected set since f is continuous in these regions. Any ball whose closure lies in the strip -1 g_x < 0 has its closure mapped onto [0}. Any ball which lies to the left Of and is tangent to x = -1 has its closure mapped onto [-l,1]. The same is true Of any ball 'which overlaps x = -l orwx = 0. Finally' f(R) = [-1,l]. However, the inverse image of the closed set (-m,0] is not E closed since R is contained in f-1((-m,0]) but R is not. Remark. As before, let 8 be the family of all Baire 1 functions mapping En into a fixed separable metric space Y. If one combines Theorem 2.2 and Theorem 2.3, one gets the desired condition distinguishing those functions in 3 'which are Darboux (E), where E is any basis for En satisfying (1) and (2). As was remarked earlier, this condition is similar to a condition established by Neugebauer [6] and discussed by Weil [7] for Baire 1 functions having the ordinary Darboux prOperty. Let Y be a separable metric space. Let f : En 4'Y be Baire 1. Let E 'be a basis of connected sets for En satisfying (1) and (2). Then f-1(K) is E closed for all closed K c‘Y if and only if f is Darboux (E). 24 The following definition is due to Bruckner and Bruckner [2]. Definition 2.5. Let X be a tOpological space, and let E ‘be a basis for X. A set S CJX is said to be dense-in-itself (E) if, given any x 6 S and any U 6 E ‘with x E H, S n U contains some point other than x. Lemma 2.1. Let f : X 4‘Y, where X and Y are topological spaces. Let E 'be a basis for X satisfying (3). Then f-1(V) is dense-in-itself l (E) for all Open V c:Y if and only if f- (K) is E closed for all closed K C‘Y. Proof. Let f-1 (V) be dense-in—itself (E) for all Open V c:Y, but suppose there is some closed K c:Y such that f—1(K) is not E closed. Then there is some U E E and some x 6 Bd U such that 1m) and x c {10140. But f-1(Y~K) is UCf' dense—in-itself (E). Hence U H f-1(Y~K) # ¢ ‘which is a contradiction. For the reverse implication, let f-1(K) be E closed for all closed K c‘Y, but suppose there is some Open V C Y such that f-1(V) is not dense-in-itself (E). Then there is some U G E and some x e f-1(V) 25 such that x E U and U n f-1(V) c {x}. If x 6 U, then, by (3), there is a U 6 E such that U crU 1 If X £ U, then, obviously, I and x 6 Bd U1. x C Bd U. We see that in either case there is a W 6 E such that W c:U and x 6 Bd W. By the choice Of U and x we have ‘W c f_l(Y~V) which is a contradiction because f-1(Y~V) is E closed. Remark. Notice that, in Lemma 2.1, the hypothesis that E satisfies (3) was used only to show that, if f-1(K) is E closed for all closed K C'Y, then f-1(V) is dense-in-itself (E) for all open V c'Y. (The reverse implication was proved without using this assumption.) That this hypothesis cannot be omitted even if one assumes that f is Baire l is shown by the following example. Example 2.12. Let X = Y = E1. Let E be the basis for El consisting Of all Open intervals not having 0 as an endpoint. As was pointed out in Example 2.5, E does not satisfy (3). Let f be the characteristic function Of the singleton [0}. Clearly f is Baire l. The inverse image Of any set is either {0}, El ~ [0}, ¢, or E1. Hence the inverse image Of any closed set is E closed. HOwever, the inverse image Of the Open set (O,m), for example, is not dense- in-itself (E). 26 As before, let 8 be the family Of all Baire 1 functions mapping En into a fixed separable metric space Y. As was remarked earlier, Bruckner and Bruckner [2] established a condition (stated in terms Of inverse images Of Open sets) distinguishing those functions in 3 'which are Darboux (E), 'where E is any basis for En satisfying certain restrictions. This condition is similar to a condition established by Zahorski [8] for Baire 1 functions having the ordinary Darboux prOperty. Bruckner and Bruckner's result is presented here as a simple consequence Of the remark following Example 2.11. Theorem 2.4. Let Y be a separable metric space. Let f : En 4‘Y be Baire 1. Let E The a basis of connected sets for En satisfying (1) and (2). Then f-1(V) is dense-in-itself (E) for all Open V c'Y if and only if f is Darboux (E). Proof. Since E is a basis for En satisfying (1), E satisfies (3) also. Hence, by Lemma 2.1, f-1(V) is dense-in-itself (E) for all Open V C‘Y if and only if f-1(K) is E closed for all closed K c‘Y. But, by the remark following Example 2.11, f-1(K) is E closed for all closed K C'Y if and only if f is Darboux (E). 27 Remark. In the proof of Theorem 2.4, the assumption that E satisfies (2) was used only to show that, if f is Darboux (E), then f-1(V) is dense-in-itself (E) for all Open V c'Y. The assumption that E satisfies (1), however, was used for both implications. The following lemma gives an additional interesting prOperty Of Darboux (E) functions. This lemma will be used in the next chapter. Lemma 2.2. Let X be a metric space and Y a tOpOlogical space. Let E ‘be a basis for X. If f : X 4'Y is Darboux (E), then f(V) is connected for each connected Open set V CLX. Proof. Let V CIX be Open and connected, but suppose that f(V) is not connected. Then there exist sets A* and 3*, Open in Y, such that f(v) = (f(V) nA*) U (f(V) n 13*) is a decomposition Of f(V) intO two disjoint, non-void, relatively Open sets. * Let A=[x€V:f(x)€A}. Let ‘k B = {x E V : f(x) 5 B }. Notice that V = A U B and A n B = ¢. It is claimed that both A and B are Open in X. TO show that A is Open, let x 6 A. 28 Since V is Open, one can find an Open ball W such that x E W ch CIV. Since W“ is Open, one can find some UEE suchthat XGUCW. Hence xEUchV. It then follows that U ClA, for suppose not. One has U H A #’¢' since x 6 U H A. If U H B # ¢ also, then f(U) = (f(U) n A*) U (f(U) n 13*) is a decomposition Of f(U) into two disjoint, non-void, relatively Open sets. But this is impossible since f is Darboux (E). Therefore one has x 6 U CIU c4A. Hence A is Open. (The argument for B is identical.) But V = A U B, and V is connected. Therefore A = ¢ or B = ¢. Hence f(V) is connected. CHAPTER III A DARBOUX PROPERTY FOR THE GRADIENT This chapter deals with real-valued functions defined on En. It is tacitly assumed that, for each function f considered here, grad f exists everywhere. Definition 3.1. Let (X,d) be a metric space. Let S c.X and let x 6 §. Then x is said to be accessible from S if there is some a > O with the following property: there exists a sequence Of points cn 6 S ~'{x} such that lim cn = x and B(cn,dd(cn,x)) C’S for each n. Definition 3.2. Let X be a metric space. A set S C X is said to have an accessible boundary if each x c Bd S is accessible from S. Roughly speaking, a set S has an accessible boundary if it does not come to a cusp on its boundary. Consider the following example. Example 3.1. Given 5 > 0, let XS be the unique real root Of 2x3 + x - s = 0. Then 29 30 4 1/2 _ 2 r8 — [(3 x8) + xS is the minimum distance from (3,0) to the curve y = x2. The minimum distance from (5,0) to the curve y = x is --. Define US and VS as follows. J2“ { (X.y) C II \Y\ < x2,0 < x < XS] U B((s,0),rs) M < x, 0 < x < 3} U B((s.o>.-§—> 2 v {(X.y) 8 Vs has an accessible boundary, but US does not. The point (0,0) is on the boundary Of V5 and is accessible from Vs' One can choose a tO be any positive number not exceeding sin E . The point (0,0) is on the boundary Of US also but is not accessible from U . 3 Remark. Notice that, if x 6 Int S, x is automatically accessible from S. In fact, one can choose a to be 1. The following theorem is similar tO a result established by Borwein and Meier [l] for the gradient Of a function which is not necessarily differentiable. Theorem 3.1. Let E 'be a basis for En' each element Of which has an accessible boundary. Let f : Bn 4 E be differentiable, and let F = grad f. l 31 Assume that f(0) = 0 and F(0) = 0. Let U be any element of E such that 0 G U. Then, given any e 2 0, there is some v P U such that v # O and ‘F(v)} g 5. Proof. First we use the fact that U has an accessible boundary to select an a, 0 \ a g l, with the following property: given any 5 > 0, there is a point Cg € U, such that Cg # 0 and B(c§,a‘c§}) CU n B(O,§). Next we use the fact that f is differentiable, that f(0) = 0, and that F(0) = 0 to find a 6 > 0 such that \f(x)‘ < %f-‘x‘ whenever ‘x‘ 3,5. Finally select a c E U such that c # 0 and B = B(c,O‘c\) C U H B(0,6). Now let g(x) = €2\x-c\2 - f(x)2. Since 9 is continuous and E is compact, there is some v E E such that g(v) = min{g(x) : x E B}. For convenience let 1 = min[g(x) : x E E}. For x 6 Bd B one has g(x) 2 6: C12 ‘C‘Z - % a262\x‘2 2 czaz‘c‘z - Al‘ (1262(‘C‘ + o‘c‘)2 > €2a2\c\2 _ % d2e2(2‘c\)2 = O 32 But 1 g_g(c) = -f(c)2 g_0. Hence g(v) = x for some v C B. Therefore grad g(v) = 2e2(v-c) 2f(v) grad f(v) = 0. Hence f(V) grad f(v) = 32(v—c), which implies that (*) f(v>2\grad f(v)‘2 e4‘V-C|2 = e2(g(v)+f(v)2) 62 (1+f(V)2) g €2f(v)2. If 1 < 0, then f(v)2 = €21v-c12 - g(v) 2,-g 0. Hence, dividing (*) by f(v)2 yields the desired result. Similarly, if 1 = 0 and there is some v 6 B such that V # c and g(v) = 1, then one has f(v)2 = Again, dividing (*) by f(v)2 yields the desired result. It remains to consider the case where 1 = 0 and g(x) > 0 for all x C B except x = c. (Notice that, in this case, g(c) = f(c) = 0.) Let B1 = B(c,%‘c\). For x 6 Bd Bl one has f(x)2 = % czaz‘c\2 - g(x) < % €2a2 c‘z. Hence |f(x)‘ < % am\c\ for all x 6 Bd 81' Since f is continuous and Bd B is compact, there is some u 4 Bd B 1 1 33 : X 6 Bd Bl}. v > 0 such such that f(u) = max{f(x) Since f(u) < % eo‘c‘, one can choose a that f(u) + v < % ga‘c‘. Then, for all x 6 Bd B1, one has -% €C1‘C‘< f(x) < f(x) + vgf(u) + V < % €a|c\. Hence, for all x 6 Bd Bl' \f(x)+v‘ < % ea‘c‘. Now let €2\x-c\2 - (f(X)+v)2. h(x) = is compact, there is Since h is continuous and B1 some v C E1 such that h(V) = min{h(x) x E El} For convenience let u = min[h(x) : x 6 El}. For x G Bd Bl one has 1 2 2 h(X) / Z EZUZ‘C‘Z - 211- u e c‘ = 0 But u gh(c) = -(f(c)+\2)2 = -v2 \ 0. Hence h(v) = u for some v 6 81' Therefore 262(V-C) - 2(f(v)+v) grad f(v) = 0. grad h(v) — which implies that Hence (f(v)+v) grad f(V) €2(V-C), (**> (f(V)+v)2}grad f(v)\2 = e4‘v-c\2 52 (h(V)+(f(V)+v) 2) .301 + (f(vmnz) g 62(f(v)+v)2. 34 But (f(v)+v)2 = e2\v-c\2 - h(v) = e2\v-c\2 - u 2-u >0. Hence, dividing (**) by (f(v)+v)2 yields the desired result. Corollary 3.1. Let E ‘be a basis for En' each element Of which has an accessible boundary. Let f : Bn 4 E be differentiable, and let F = grad f. 1 Let U be any element of E, and let x0 be any point in U. Then, given any 9 > 0, there is some v E U such that v # x0 and ‘F(v)-F(xo)‘ 3.5. Proof. Let G(x) = F(x+xo) - F(xo), and let g(x) = f(x+xo) — f(xo) - F(xo) 0 x, where 0 denotes the usual scalar product in En' It is clear that g is differentiable, G = grad g, g(0) = 0, and G(0) = 0. Let E1 = {—xO+W : W P E]. E1 is a baSis for En' each element of which has an accessible boundary. Moreover, V = -xo+U is in E1, and 0 E V. Hence, by Theorem 3.1, there is some v' 6 V such that v’ # 0 I and ‘G(v')| g e. Let v = v + x0. Then v 6 U, v #’x0, and one has \F(v)-F(xo)‘ = lF(v’+xO)-F(x0)| = ‘G(v')\ g e. 35 The following theorem is the major result in this chapter. Theorem 3.2. Let E be a basis of connected sets for En' each element Of which has an accessible boundary. Let f : En 4 E1 be differentiable, and let F = grad f. Then F is Darboux (E). Proof. Let Eb be the Open-ball basis for En' Let E1 = E U E0. Then E1 is a ba31s for En satisfying (1), and each element of E1 is a connected set with an accessible boundary. Note that F, being the gradient of a continuous function, is Baire 1. Suppose F is not Darboux (El). Then, by Theorem 2.4, there is an Open set V c En such that F-1(V) is not dense-in-itself (E1). Hence there is some U 9 El and some xo 6 U n F-1(V) such that U contains no point of F-1(V) with the possible exception of x . O We have F(XO) E V, and V is Open. Hence there is some r > 0 such that B(F(xo),2r) c‘V. Now use Corollary 3.1 to find a v G U such that v #'x0 and ‘F(v)—F(xo)‘ g_r. But then F(v) E B(F(xo),2r) c:V which is a contradiction. Thus F must have been Darboux (E1) and, hence, Darboux (E). 36 A Darboux property for partial derivatives follows easily from Theorem 3.2. Corollary 3.2. Let E be a basis Of connected sets for En' each element Of which has an accessible boundary. Let f : En 4 E be differentiable. Then 1 akf (the partial derivative with respect to the kth variable) is Darboux (E) for k = l,...,n. PrOOf. Let pk : Bn 4 E1 be the projection onto the kth component. Then akf(x) = pk(grad f(x)). Since, as is well-known, pk is continuous and, by Theorem 3.2, grad f is Darboux (E), it follows immediately that akf is Darboux (E). The following corollary establishes a sort Of intermediate value property for the gradient Of a differentiable function. Corollary 3.3. Let E be a basis Of connected sets for En' each element Of which has an accessible boundary. Let F : En 4 En be the gradient of a differentiable function. Suppose that ‘F(x)| assumes the values a and b, a < b, at the points u and v respectively. Then, if U is any element Of E such that u,v 6 U, \F(x)} assumes every value between a and b in U. 37 Proof. F is Darboux (E) Tby Theorem 3.2. Thus F(U) is connected by Lemma 2.2. Hence [\F(w)‘ : w E U} is an interval. Now suppose there is some c 6 (a,b) with the following property: there exists no w E U such that ‘F(w)‘ = c. Let s = % min[c-a,b-c}. By Corollary 3.1, one can find points w .w2 e U such 1 that ‘F(wl)-F(u)‘ g_e and ‘F(w2)—F(v)“g 6. Hence ”F(wl) |-a‘ = HF(W1)\ - (F(u) H _<_ lF(wl)-F(u)\ g e. and HF 0, let US be as defined in Example 3.1. The collection Of all such US together with translations of same is a basis for the usual tOpOlogy on E Call this basis E. 2. Each U 6 E is connected, but each U E E has a point on its boundary which is not accessible from U. Let 5 -4 3 l4 -2 2 32 h(XaY)='2—7X y --§-x y +—9-y— b) N 2 x N 7 for x > 0. Define f : E 4 E as follows. ‘y‘ 24x2 and X > 0 O -h(x,-y) , —4x < y < -x2 and x > 0 f(X.Y) = y , ‘y‘ g_x2 and x > 0 h(x,y) , x \ y < 4x and x > 0 0 , X g_0 at is an elementary though rather tedious task to show that f is indeed everywhere differentiable.) NOw let F = grad f. We have F(0,0) = (0,0). However, F maps the set {(x,y) : \yl ; X2, x > 0} onto the point (0,1). Hence, for each s > 0, F maps US onto a set which is not connected. Therefore F is not Darboux (E). The hypothesis in Theorem 3.2 that f be differentiable cannot be omitted as the following example shows. 39 Example 3.3. Let E be any basis whatever for E2. Let 2xy(x2+y2)-l/2. (X.Y) # (0,0) f(x,y) = O I (XIY) = (0,0). Then ’12Y3(X2+Y2)—3/2. 2x3(x2+y2)_3/2). (X.y) # (0.0) grad f(X.y) = (000) I (XIY) = (0:0)- For (X:y) ¢ (0.0). ‘grad f(x,y)}2 = 4(X6+Y6)(X2+y2)-3 + cos 49‘; l, I NIU'I le where x = r cos 6 and y = r sin 9. But ‘grad f(0,0)‘ = 0. Thus, for any Open neighborhood U Of (0,0), grad f(U) is not connected. The function f is differentiable everywhere save at the origin, for f,fx, and fy are continuous everywhere save at the origin. But grad f is certainly not Darboux (E). Remark. Let E be any basis of connected sets for En' Let f : ED 4 El be continuously differentiable, and let F = grad f. Then F is clearly Darboux (E). BIBLIOGRAPHY BIBLIOGRAPHY D. Borwein and A. Meier, A prOperty Of gradients, Amer. Math. MOnthly 76 (1969), pp.648—649. A.M. Bruckner and J.B. Bruckner, Darboux transformations, Trans. Amer. Math. Soc. 128 (1967). pp.lO3-111. A.M. Bruckner and J.G. Ceder, Darboux continuity, Jber. Deutsch Math.-Verein 67 (1965), pp.93-ll7. C. Kuratowski, Topologie I, 4th ed., Monogr. Mat. VOl. 20, Warsaw, 1958. L. Mi§ik, Uber die Funktionen der ersten Baireschen Klasse mit der Eigenschaft von Darboux, Mat.-Fyz. éasopis Sloven. Akad. vied 14 (1964). pp.44-49. C.J. Neugebauer, Darboux functions Of Baire class one and derivatives, Proc. Amer. Math. Soc. 13 (1962), pp.838-843. C.E. Weil, A tOpOlogical lemma and applications to real functions, Pacific J. Math., to appear. Z. Zahorski, Sur la premiere dérivée, Trans. Amer. Math. Soc. 69 (1950), pp.l—54. 4O