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 ( x ) = ∑ n = 1 ∞ b n d ( x , S n ) , where S n is a maximal 1 a n -separated set in a metric space X . So, if X = R and S n = 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 Lip f a , b ( x ) = + ∞ if a > b > 2 and x is a non-isolated point of X . Moreover, if the shell porosity p s ( X , x ) < λ < 1 for some λ and each non-isolated point x ∈ X then the little Lipschitz derivative lip f a , b ( x ) = + ∞ for large enough a > b and any non-isolated point x ∈ 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 Lip f ( x ) = + ∞ (and lip f ( x ) = + ∞ ) exactly on A .