Takagi-van der Waerden functions in metric spaces and its Lipschitz derivatives

TL;DR

This paper introduces a generalized Takagi-van der Waerden function in metric spaces, analyzing its Lipschitz derivatives and demonstrating conditions under which these derivatives are infinite, extending classical nowhere differentiable functions.

Contribution

It generalizes Takagi and van der Waerden functions to metric spaces and characterizes their Lipschitz derivatives, including conditions for infinite derivatives at non-isolated points.

Findings

Lipschitz derivative is infinite at non-isolated points for certain parameters.

Conditions involving shell porosity imply infinite little Lipschitz derivatives.

Existence of continuous functions with infinite Lipschitz derivatives on arbitrary open sets.

Abstract

We introduce the Takagi--van der Waerden function with parameters $a{>}b{>}0$ by setting $f_{a,b}(x)=\sum\limits_{n=1}^\infty b^n d\big(x,S_n\big)$, where $S_n$ is a maximal $\frac1{a^n}$-separated set in a metric space $X$. So, if $X=\mathbb R$ and $S_n=\frac1{a^n}\mathbb Z$ then $f_{2,1}$ is the Takagi function and $f_{10,1}$ is the van der Waerden function which are the famous examples of nowhere differentiable functions. Then we prove that the big Lipschitz derivative $\mathrm{Lip} f_{a,b}(x)=+\infty$ if $a>b>2$ and $x$ is a non-isolated point of $X$. Moreover, if the shell porosity $p^s(X,x)<\lambda<1$ for some $\lambda$ and each non-isolated point $x\in X$ then the little Lipschitz derivative $\mathrm{lip} f_{a,b}(x)=+\infty$ for large enough $a>b$ and any non-isolated point $x\in X$. In particular, this is true for any normed space. Finally, we prove that for any open set $A$ in a metric (normed) space $X$ without isolated points there exists a continuous function $f$ such that $\mathrm{Lip} f(x)=+\infty$ (and $\mathrm{lip} f(x)=+\infty$) exactly on $A$.