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

We introduce the Takagi-van der Waerden function with parameters a>b>0 by setting fa,b(x)=∑ₙ₌₁^∞ bⁿ d(x,Sₙ), where Sₙ is a maximal 1aⁿ-separated set in a metric space X without isolated points. So, if X= R and Sₙ=1aⁿ Z then f2,1 is the Takagi function and f10,1 is the van der Waerden function which are the famous examples of nowhere differentiable functions. Then we prove that the big Lipschitz derivative Lip fa,b=+∞ if $a>b>2$. Moreover, if X is a normed space then the little Lipschitz derivative lip fa,b=+∞ for large enough $a>b$. Thus, we prove that for any open set A in a metric (normed) space X there exists a continuous function f such that Lip f(x)=+∞ (and lip f(x)=+∞) exactly on A.