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Summary: On a flexible class of continuous functions
with uniform local structure
Pieter C. Allaart
February 18, 2010
Abstract
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.
AMS 2000 subject classification: 26A27 (primary); 26A15, 26A30,
60G50 (secondary)
Key words and phrases: Takagi function, Nowhere-differentiable
function, Modulus of continuity, Law of the iterated logarithm, Bino-
mial measure, Gray code.
1 Introduction
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