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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.

Information

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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3. Saks, S., Theory of the integral, Monografje Matematyczne, Vol. 7, Warszawa - Lwów, 1937.Google Scholar
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