A subset of metric preserving functions (Q1384266)

The subset \({\mathcal D}\) of the set \({\mathcal M}\) of metric preserving functions which is considered is defined as follows: Let \(f\in{\mathcal M}\) be differentiable on \((0,\infty)\), \[ g(x):= \begin{cases} f'(x), &\quad x\in (0,\infty),\\ 0, &\quad x=0 \end{cases} \] then \(f\in{\mathcal D}\) iff \(g\in{\mathcal M}\). The purpose of this paper is to give examples of these types of functions and to characterize the type of \(f\) which can be in \({\mathcal D}\).