SOME REMARKS ON METRIC PRESERVING FUNCTIONS

Abstract

The purpose of this paper is to study a behavior of continuous metric preserving functions $f$ with $f^{\prime }(0)=+\infty$. First we show, via a simple example, that it is possible that such a function has no finite derivatives at any point. Then in Example 2 we construct a nondecreasing, differentiable, metric preserving function having infinite derivative at least at the points $x=2^{-n}$ for each natural number, $n$.