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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
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