A new generalization of the Takagi function

TL;DR

This paper introduces a broad class of functions generalizing the Takagi function, analyzing their real-analytic properties, including differentiability, Hausdorff dimension, and local monotonicity, extending known results for the original Takagi function.

Contribution

It defines a new family of functions parameterized by matrices that generalize the Takagi function and studies their detailed analytical properties.

Findings

Includes the original Takagi function as a special case.

Analyzes Hausdorff dimension of the graph of these functions.

Establishes differentiability and modulus of continuity results.

Abstract

We consider a one-parameter family of functions $\{F(t,x)\}_{t}$ on $[0,1]$ and partial derivatives $\partial_{t}^{k} F(t, x)$ with respect to the parameter $t$. Each function of the class is defined by a certain pair of two square matrices of order two. The class includes the Lebesgue singular functions and other singular functions. Our approach to the Takagi function is similar to Hata and Yamaguti. The class of partial derivatives $\partial_{t}^{k} F(t, x)$ includes the original Takagi function and some generalizations. We consider real-analytic properties of $\partial_{t}^{k} F(t, x)$ as a function of $x$, specifically, differentiability, the Hausdorff dimension of the graph, the asymptotic around dyadic rationals, variation, a question of local monotonicity and a modulus of continuity. Our results are extensions of some results for the original Takagi function and some generalizations.