Strongly ultrametric preserving functions

An ultrametric preserving function $f$ is said to be strongly ultrametric preserving if ultrametrics $d$ and $f\circ d$ define the same topology on $X$ for each ultrametric space $(X,d)$. The set of all strongly ultrametric preserving functions is characterized by several distinctive features. In particular, it is shown that an ultrametric preserving $f$ belongs to this set iff $f$ preserves the property to be compact.