## Real Analysis Exchange

### The Oscillation Function on Metric Spaces

Zbigniew Duszyński

#### 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 $y\in [0,1]$.

#### Article information

Source
Real Anal. Exchange, Volume 25, Number 1 (1999), 489-492.

Dates
First available in Project Euclid: 5 January 2009

https://projecteuclid.org/euclid.rae/1231187624

Mathematical Reviews number (MathSciNet)
MR1758906

Zentralblatt MATH identifier
1015.54010

#### Citation

Duszyński, Zbigniew. The Oscillation Function on Metric Spaces. Real Anal. Exchange 25 (1999), no. 1, 489--492. https://projecteuclid.org/euclid.rae/1231187624

#### References

• Z. Duszyński, On the function of oscillation, Problemy Matematyczne 14, Bydgoszcz 1995, 15–20.
• R. Sikorski, Funkcje rzeczywiste, PWN, Warszawa, 1958