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1999/2000 The Oscillation Function on Metric Spaces
Zbigniew Duszyński
Real Anal. Exchange 25(1): 489-492 (1999/2000).


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 $y\in [0,1]$.


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Zbigniew Duszyński. "The Oscillation Function on Metric Spaces." Real Anal. Exchange 25 (1) 489 - 492, 1999/2000.


Published: 1999/2000
First available in Project Euclid: 5 January 2009

zbMATH: 1015.54010
MathSciNet: MR1758906

Primary: 26A15

Keywords: oscillation function , separable metric space

Rights: Copyright © 1999 Michigan State University Press

Vol.25 • No. 1 • 1999/2000
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