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 f₀, ₁ (x) =₍=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 f₀, ₁=+ if a>b>2. Moreover, if X is a normed space then the little Lipschitz derivative lip f₀, ₁=+ 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.