Journal of the Mathematical Society of Japan

On a flexible class of continuous functions with uniform local structure


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This paper considers a class of continuous functions constructed as a series of iterates of the “tent map” multiplied by variable signs. This class includes Takagi's nowhere-differentiable function, and contains the functions studied by Hata and Yamaguti [Japan J. Appl. Math., 1 (1984), 183-199] and Kono [Acta Math. Hungar., 49 (1987), 315-324] as a proper subclass. A complete description is given of the differentiability properties of the functions in this class, and several statements are proved concerning their uniform and local moduli of continuity. The results are applied to generation of random functions.

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J. Math. Soc. Japan Volume 61, Number 1 (2009), 237-262.

First available in Project Euclid: 9 February 2009

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Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
Secondary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 26A30: Singular functions, Cantor functions, functions with other special properties 60G50: Sums of independent random variables; random walks

Takagi function nowhere-differentiable function modulus of continuity law of the iterated logarithm binomial measure Gray code


ALLAART, Pieter C. On a flexible class of continuous functions with uniform local structure. J. Math. Soc. Japan 61 (2009), no. 1, 237--262. doi:10.2969/jmsj/06110237.

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