Real Analysis Exchange

The Oscillation Function on Metric Spaces

Zbigniew Duszyński

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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

Permanent link to this document
https://projecteuclid.org/euclid.rae/1231187624

Mathematical Reviews number (MathSciNet)
MR1758906

Zentralblatt MATH identifier
1015.54010

Subjects
Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}

Keywords
oscillation function separable metric space

Citation

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


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References

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