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

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 $A$.