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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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

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

First available in Project Euclid: 2 October 2009

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Zentralblatt MATH identifier

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

Continuous non-differentiable function the law of the iterated logarithm


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.

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