Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the critical case of Okamoto’s continuous non-differentiable functions

Kenta Kobayashi

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Abstract

In a recent paper in this Proceedings, H. Okamoto presented a parameterized family of continuous functions which contains Bourbaki’s and Perkins’s nowhere differentiable functions as well as the Cantor-Lebesgue singular function. He showed that the function changes it’s differentiability from ‘differentiable almost everywhere’ to ‘non-differentiable almost everywhere’ at a certain parameter value. However, differentiability of the function at the critical parameter value remained unknown. For this problem, we prove that the function is non-differentiable almost everywhere at the critical case.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 85, Number 8 (2009), 101-104.

Dates
First available in Project Euclid: 2 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.pja/1254491212

Digital Object Identifier
doi:10.3792/pjaa.85.101

Mathematical Reviews number (MathSciNet)
MR2561897

Zentralblatt MATH identifier
1184.26003

Subjects
Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
Secondary: 26A30: Singular functions, Cantor functions, functions with other special properties

Keywords
Continuous non-differentiable function the law of the iterated logarithm

Citation

Kobayashi, Kenta. On the critical case of Okamoto’s continuous non-differentiable functions. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 8, 101--104. doi:10.3792/pjaa.85.101. https://projecteuclid.org/euclid.pja/1254491212


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References

  • N. Bourbaki, Functions of a real variable, Translated from the 1976 French original by Philip Spain, Springer, Berlin, 2004.
  • P. Hartman and A. Wintner, On the law of the iterated logarithm, Amer. J. Math. 63 (1941), no. 1, 169–176.
  • H. Okamoto, A remark on continuous, nowhere differentiable functions, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 3, 47–50.
  • H. Okamoto and M. Wunsch, A geometric construction of continuous, strictly increasing singular functions, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 7, 114–118.
  • F. W. Perkins, An Elementary Example of a Continuous Non-Differentiable Function, Amer. Math. Monthly 34 (1927), no. 9, 476–478.