The Oscillation Function on Metric Spaces

Abstract

For each metric space $(X,\xi)$ and each bounded function $f\colon X\to \R$ the family of the sets $\O_f(y)=\{x\in X\colon\, \om_f(x)\ge y\}$ ($\om_f(x)$ is the oscillation of $f$) has some well known properties. In this paper it is constructively shown that for each family $\{\O(y)\}_{y\in [0,1]}$ of subsets of $X$ (separable and $\C$-dense in itself) having similar properties there exists a function $f\colon\, X\to [0,1]$ such that $\O_f(y)=\O(y)$ for each $y\in [0,1]$.