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On Differentiable Functions having an Everywhere Dense set of Intervals of Constancy

Published online by Cambridge University Press:  20 November 2018

A. M. Bruckner  and
John L. Leonard
A. M. Bruckner
Affiliation:
University of California Santa Barbara
John L. Leonard
Affiliation:
University of California Santa Barbara
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The Cantor function C [2; p. 213], which appears in analysis as a simple example of a continuous increasing function which is not absolutely continuous, has the following properties:

  1. (i) C is defined on [0,1], with C(0) = 0, C (l) = l;

  2. (ii) C is continuous and non-decreasing on [0,1];

  3. (iii) C is constant on each interval contiguous to the perfect Cantor set P;

  4. (iv) C fails to be constant on any open interval containing points of P;

  5. (v) The set of points at which C is non-differentiable is non-denumerable.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 8 , Issue 1 , February 1965 , pp. 73 - 76
Copyright
Copyright © Canadian Mathematical Society 1965

References

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2. Natanson, I. P., Theory of functions of a real variable, Vol. 1, Ungar, New York, 1961.Google Scholar
3. Saks, S., Theory of the integral, Monografje Matematyczne, Vol. 7, Warszawa - Lwów, 1937.Google Scholar
4. Zahorski, Z., Sur la première dérivée, Trans. Amer. Math. Soc. 69 (1950), 1-54.Google Scholar
5. Zahorski, Z., Űber die Konstruktion einer differenzierbaren, monotonen, nicht Konstanten Funktion, mit űberall dichter Menge von Konstanzintervallen, Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, classe III, vol 30(1937), 202-206.Google Scholar