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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.

##### MSC:
 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
##### Keywords:
metric preserving functions