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On modifications of the Euclidean metric on reals. (English) Zbl 0928.26008
Denote by \(\mathcal O\) the set of all functions \(f: [0,\infty) \to [0,\infty)\) with \(f(0)=0\). Let \((M,d)\) be a metric space. For each \(f\in \mathcal O\) define the function \(d_f: M\times M \to [0,\infty)\) as follows \[ d_f(x,y)=f\bigl (d(x,y)\bigr)\quad \text{for each } x,y \in M. \] Denote by \(\mathcal M\) the set of all functions \(f\in \mathcal O\) such that for each metric space \((M,d)\) the function \(d_f\) is a metric on \(M\). Denote by \(\mathcal M_0 (\mathcal M_1)\) the set of all functions \(f \in \mathcal O\) such that \(e_f\) is a pseudometric (metric) on the real line, where \(e: \mathbb R \times \mathbb R \to [0,\infty)\) is the Euclidean metric on \(\mathbb R\).
It is shown that if \(f \in \mathcal M_0\) is nonconstant and \(f\) is differentiable on \((0,\infty)\), then \(f\in \mathcal M_1\).
Also a construction of a differentiable function \(f \in \mathcal M_1\) with \(\liminf _ {x\to \infty }f(x)=0\) is described.

26B35 Special properties of functions of several variables, Hölder conditions, etc.
26A30 Singular functions, Cantor functions, functions with other special properties
54E40 Special maps on metric spaces