Takagi–van der Waerden functions in metric spaces and their Lipschitz derivatives

We introduce the Takagi–van der Waerden function with parameters $a>b>0$ by setting $f_{a,b}\left(x\right)=\sum _{n=1}^{\infty }{b}^{n}d(x,S_n)$, where $S_n$ is a maximal $\frac{1}{a^n}$-separated set in a metric space X. So, if $X=R$ and $S_n=\frac{1}{a^n}Z$ then $f_{2,1}$ is the Takagi function and $f_{10,1}$ is the van der Waerden function which are well-known examples of nowhere differentiable functions. Then we prove that the big Lipschitz derivative $\mathrm{Lip}{f}_{a,b}\left(x\right)=+\infty$ 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 λ and each non-isolated point $x\in X$ then the little Lipschitz derivative $\mathrm{lip}{f}_{a,b}\left(x\right)=+\infty$ 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 $\mathrm{Lip}f\left(x\right)=+\infty$ (and $\mathrm{lip}f\left(x\right)=+\infty$) exactly on A.