Real Analysis Exchange

The Takagi Function: a Survey

Pieter C. Allaart and Kiko Kawamura

Full-text: Open access

Abstract

This paper sketches the history of the Takagi function T and surveys known properties of T, including its nowhere-differentiability, modulus of continuity, graphical properties and level sets. Several generalizations of the Takagi function, in as far as they are based on the tent map, are also discussed. The final section reviews a number of applications of the Takagi function to various areas of mathematics, including number theory, combinatorics and classical real analysis.

Article information

Source
Real Anal. Exchange, Volume 37, Number 1 (2011), 1-54.

Dates
First available in Project Euclid: 30 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1335806762

Mathematical Reviews number (MathSciNet)
MR3016850

Zentralblatt MATH identifier
1248.26007

Subjects
Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
Secondary: 26A16: Lipschitz (Hölder) classes 26A78 26A80

Keywords
Takagi function van der Waerden function continuous nowhere differentiable function modulus of continuity infinite derivatives level set Hausdorff dimension Lebesgue's singular function functional equation

Citation

Allaart, Pieter C.; Kawamura, Kiko. The Takagi Function: a Survey. Real Anal. Exchange 37 (2011), no. 1, 1--54. https://projecteuclid.org/euclid.rae/1335806762


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