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Some Properties of Almost Everywhere Non - Differentiable Functions

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Some Properties of Almost Everywhere Non - Differentiable Functions DEMONSTRAHO MATHEMATICA Vol. XVI No 3 1983 Bozena Szkopinska, Janusz Jaskula SOME PROPERTIES OF ALMOST EVERYWHERE NON - DIFFERENTIABLE FUNCTIONS In the paper we shall apply the following notation: Af = {x: | f (x)|<oo} , AfW = {x: f (*) = +00}, Df - the set of all points of discontinuity of a func- tion f, Cf - the set of all points of continuity of the func- tion f, Mf - the set of all those points at which there is no one-sided (finite or infinite) derivative, I - an arbitrary olosed interval with finite Lebesgue measure, R - the Bet of all real numbers, |a| - the Lebesgue measure of a set A, f(x) - the lower derivative of the function f at a point x, f(x) - the upper derivative' of the function f at the point x, A - the closure of the set A. Theorem. If a. set Be Gg, |b| = 0, and there exists a function f s I—R satisfying the conditionsi a) Af(oo) = B b) |l\Mf| = 0, then the set B is nowhere dense. - 695 - 2 B. Szkopinska, J. Jaskuia Proof. It follows from the theorem of Denjoy-Young - -Saks [3] that almost each point x of non-differentiability of a function, in particular, almost each point of the set Mf, is a point at which (1) f(x) = -00 and f(x) = +00. Henoe and from assumption b) it follows that almost each point of the interval I has property (1). This means that the sets {x: f(x) = -00} and {x: f(x) = +00} are dense in I. Whenoe, in virtue of Lemma, [1] p.74, these sets are residual. Consequent- ly, such is the set E = {11 fix) = -oo}n{x: f(x) = +00}. Suppose that the set B is not nowhere dense. Then there exists an interval (a,b) cI on which the set B is dense. Since BeGg, the set Bn(a,b) is residual on (a,b). The set EnBn n (a,b) is therefore residual in (a,b). This last sentenoe is false because the set EnBn(a,b) is, of course, empty. Corollary 1. If a function f is continuous, f:I —i• R, and a set B = Af{oo) and I\B = Mf, then B is no- where dense. Corollary 2. For an arbitrary set B e G5, |B| = 0, B = I, there exists no function f:I -*• R such that (2) Afioo) = B, (3) Cf - Af(°°) uMf. Proof. Suppose that there exists a function f satisfying the conditions given in Corollary 2. We shall show that then (4) |Df| = 0. Let A stand for the set of those points x of the inter- val I at which the function f possesses at least one of the one-sided derivatives, and if two - they are distinot finite or infinite. Note that (5) DF c A. - 696 - Hon-differentiable functions 3 It follows from the theorem of Denjoy-Young-Saks that |A| = 0, which, along with (5), gives (4). Prom the assumptions of Corollary 2 and from (4) we get 0 = |Df| = |l\Cf| = |l\(Af(oo)uMf)| = |l\(BuMf)| =|l\Mf|, which leads to a contradiction with our theorem. Corollary 2 is a negative answer to the following question raised by Marcus in [2] s Does there exist, for any dense set B of measure zero, Be Gg , a function of type a for which A~(oo) = B and C- = ¿.(ooJulL? REFERENCES [1] M. F i 1 i p c z a k : On the derivative of a disconti- nuous function, Colloq. Math. 13 (1964). [2] S. Marcus: Sur les propriétés différentielles des fonctions dont les points de continuité forment un ensemble frontière partout dense, Ann. Sci. école Norm. Sup. (3) 79 (1958) 1-21. [3] R. S i k 0 r s k i t Funkoje rzeczywiste. T.1, Warsza- wa 1958. INSTITUTE OF MATHEMATICS, UNIVERSITY OF LÔDÎ, 90-238 iÔDÎ Received December 7, 1981. - 697 - .
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