# The Takagi function: a survey

@article{Allaart2011TheTF,
title={The Takagi function: a survey},
author={Pieter C. Allaart and Kiko Kawamura},
journal={Real analysis exchange},
year={2011},
volume={37},
pages={1-54}
}
• Published 8 October 2011
• Mathematics
• Real analysis exchange
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…
105 Citations

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(where, throughout this paper, R, Z, and N denote the sets of real numbers, integers, and positive integers, respectively, N0 = N∪{0}, and dist(x,Z) = inf{ |x−s| : s ∈ Z }). Functions of this type
Let T be Takagi’s continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire
Let T be Takagi's continuous but nowhere-differentiable function. It is known that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. We show that the most common
We classify binary self-similar sets, which are compact sets determined by two contractions on the plane, into four classes from the viewpoint of functional equations. In this classification, we can
In this paper we derive functional relations for Takagi’s continuous nowhere differentiable function T , and we give an explicit representation of T at dyadic points. As application of these
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It is well known that the number v is transcendental; the proof is complicated. However, there are several elementary proofs that X and v 2 are irrational; see for example [1]. Using a simple trick,
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The Takagi function {\tau} : [0, 1] \rightarrow [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. This paper studies the level sets L(y) = {x : {\tau}(x) = y} of the
almost everywhere, may be constant in every interval contiguous to a perfect set of measure zero: it is usually said, in this case, that f(x) is of the Cantor type. There are, however, monotonic