The Takagi function: a survey Pieter C. Allaart and Kiko Kawamura ∗ November 8, 2011 1 Introduction More than a century has passed since Takagi [75] published his simple example of a continuous but nowhere differentiable function, yet Takagi’s function – as it is now commonly referred to despite repeated rediscovery by mathematicians in the West – continues to inspire, fascinate and puzzle researchers as never before. For this reason, and also because we have noticed that many aspects of the Takagi function continue to be rediscovered with alarming frequency, we feel the time has come for a comprehensive review of the literature. Our goal is not only to give an overview of the history and known characteristics of the function, but also to discuss some of the fascinating applications it has found – some quite recently! – in such diverse areas of mathematics as number theory, combinatorics, and analysis. We also include a section on generalizations and variations of the Takagi function. In view of the overwhelming amount of literature, however, we have chosen to limit ourselves to functions based on the “tent map”. In particular, this paper shall not make more than a passing mention of the Weierstrass function and is not intended as a general overview of continuous nowhere-differentiable functions. We thank Prof. arXiv:1110.1691v2 [math.CA] 7 Nov 2011 Paul Humke for encouraging us to write this survey, and for issuing periodic cheerful reminders. 1.1 Early history Takagi’s function is indeed simple: in modern notation, it is defined by ∞ 1 T (x)= φ(2nx), (1.1) 2n n=0 X ∗Address: Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX 76203-5017, USA; E-mail:
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[email protected] 1 0.50 0.25 0 0.25 0.50 0.75 1.00 Figure 1: Graph of the Takagi function where φ(x) = dist(x, Z), the distance from x to the nearest integer.