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

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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. © 2009 The Japan Academy.

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