Centered Takagi--van der Waerden functions and their Lipschitz derivatives

Using a modification of a generalized Takagi-van der Waerden function on a metric space we prove that for any closed subset of a metric space without isolated points there exists a continuous function such that its big and local Lipschitz derivatives are equal to infinity exactly on this set. Moreover, if given space is hermetic (for example, if it is normed) then the little Lipschitz derivative has the same property.