Some Properties of Almost Everywhere Non - Differentiable Functions

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 - .

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    4 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us