# zbMATH — the first resource for mathematics

A note on metric preserving functions. (English) Zbl 0840.26006
The function $$f: R^+\to R^+$$ is called metric preserving if the function $$f\circ d: M\times M\to R^+$$ is a metric for every metric $$d: M\times M\to R^+$$, where $$(M, d)$$ is an arbitrary metric space and $$R^+$$ denotes the set of all non-negative real numbers. Every concave function $$f: R^+\to R^+$$ such that $$f(x)= 0\Leftrightarrow x= 0$$ is metric preserving. If $$f$$ is a metric preserving function, then the metric $$d_f$$ on the set $$R$$ of all real numbers defined by $d_f(x, y)= f(|x- y|)\qquad\text{for each}\quad x, y\in R$ is called modification of the Euclidean metric on $$R$$. In the paper, the two following properties of a metric space $$(R, d_f)$$ are discussed:
(A) In $$(R, d_f)$$ there is a monotone sequence of closed balls with empty intersection.
(B) For each compact set $$K$$ in $$(R, d_f)$$ there exists a closed ball $$S$$ and a compact set $$L$$ such that $$K\subseteq R\backslash S\subseteq L$$.
(B) implies (A), but (A) does not imply (B). The space $$(R, d_f)$$ has the property (B) if the function $$f$$ has the following property:
(C) There exist non-increasing functions $$g, h: R^+\to R^+$$ that are constant in no neighborhood of $$+\infty$$ such that $$\lim_{n\to \infty} g(x)= \lim_{n\to \infty} h(x)$$ and $$g(x)\leq f(x)\leq h(x)$$ in some neighborhood of $$+\infty$$.
However, there exists a metric preserving function $$f$$ without the property (C) and such that $$(R, d_f)$$ yields the property (B).
Reviewer: M.Jůza (Opava)

##### MSC:
 26A30 Singular functions, Cantor functions, functions with other special properties
##### Keywords:
metric preserving functions
Full Text: