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

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    Mathematical Reviews (MathSciNet): MR2013000
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    Mathematical Reviews (MathSciNet): MR3497
    Digital Object Identifier: doi:10.2307/2371287
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    Mathematical Reviews (MathSciNet): MR2128931
    Digital Object Identifier: doi:10.3792/pjaa.81.47
    Project Euclid: euclid.pja/1116442036
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    Mathematical Reviews (MathSciNet): MR2361422
    Digital Object Identifier: doi:10.3792/pjaa.83.114
    Project Euclid: euclid.pja/1200672011
  • F. W. Perkins, An Elementary Example of a Continuous Non-Differentiable Function, Amer. Math. Monthly 34 (1927), no. 9, 476–478.
    Mathematical Reviews (MathSciNet): MR1521291
    Digital Object Identifier: doi:10.2307/2300224

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Proceedings of the Japan Academy, Series A, Mathematical Sciences

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