On metric preserving functions (Q1825466)

The authors call a function f: \({\mathbb{R}}^+\to {\mathbb{R}}^+\) metric preserving iff the composition \(f\circ d: M\supset M\to {\mathbb{R}}^+\) is a metric for every metric d: \(M\times M\to {\mathbb{R}}^+\) on an arbitrary space M. They extend their own work in Math. Slovaca 31, 3-12 (1981; Zbl 0482.54021) and that of \textit{F. Terpe} in Proc. Conf. Topology and Measure IV, Greifswald, 189-197 (1984; Zbl 0611.54021). The new material here consists of two fairly complicated examples of metric preserving functions and a theorem which says: if f is metric preserving, then f'(0) exists (either finite or infinite) and \[ f\{0)=\inf \{k>0:\quad f(x)\leq kx\quad for\quad each\quad x\in {\mathbb{R}}^+\}. \] The authors provide a summary of their earlier work and that of Terpe.