On probabilistic properties of Takagi's function
Takagi's nowhere differentiable continuous function (rediscovered by van der Waerden) \(f: {\mathbb{R}}\to {\mathbb{R}}\) is defined by \(f(x)=\sum^{\infty}_{n=1}2^{-n}\phi^{(n)}(x),\) where \(\phi (x)=2 dist(x,{\mathbb{Z}})\) and \(\phi^{(n)}\) represents the n-th iterate of \(\phi\). The author proves the following formula: \[ \lim_{h\downarrow 0}mes\{x\in (0,1):\;\frac{f(x+h)-f(x)}{h\sqrt{\log_ 2(1/h)}}<y\}=\frac{1}{\sqrt{2\pi}}\int^{y}_{-\infty}e^{-u^ 2/2}du. \] Previously, the author proved the same result for the Weierstrass' function [Mat. Zametki 36, No.1, 35-38 (1984; Zbl 0543.42007)].
Takagi's nowhere differentiable continuous function
scientific article; zbMATH DE number 4212352