## Real Analysis Exchange

### An Oscillation Functions on the Real Line

Zbigniew Duszyński

#### Abstract

By means of a certain well known family $\mathcal {B}$ of subsets of $\mathbb{R}$ fulfilling two conditions we introduce some topologies on $\mathbb{R}$ (in Section 2 we consider the density topology). We observe that the family of the sets $\Omega _f (y):= \left\{ x \in \mathbb{R};\; \; \omega _f(x) \geq y \right\}$ for an arbitrary bounded function $f\:\mathbb{R} \to \mathbb{R}$ (where $\omega _f(x)$ is a kind of ${\cal B}$-oscillation of $f$) has three properties. Then we show that for each family $\left\{ \Omega (y) \right\} _{y \in [0,1]} \subset 2 ^{\mathbb{R}}$ having similar properties and in addition fulfilling conditions $M _1$ and $\mathcal {U}^\prime$ (known from the literature) there is a function $f: \mathbb{R} \to [0,1]$ such that $\Omega _f(y) = \Omega (y)$ for each $y \in[0,1]$. In Section 2 we prove some analogous result for the density topology.

#### Article information

Source
Real Anal. Exchange, Volume 26, Number 1 (2000), 237-244.

Dates
First available in Project Euclid: 2 January 2009

https://projecteuclid.org/euclid.rae/1230939156

Mathematical Reviews number (MathSciNet)
MR1825506

Zentralblatt MATH identifier
1015.54010

#### Citation

Duszyński, Zbigniew. An Oscillation Functions on the Real Line. Real Anal. Exchange 26 (2000), no. 1, 237--244. https://projecteuclid.org/euclid.rae/1230939156

#### References

• C. Goffman, C. Neugebauer, T. Nishiura, Density topology and approximate continuity, Duke Math J., 28 (1961), 497–506.
• J. M. Jędrzejewski, O granicy i ciągłości uogólnionej, Zeszyty Naukowe Uniwersytetu \Lódzkiego,Seria II 52 (1973), 19–38.
• J. M. Jędrzejewski, On limit numbers of real functions, Fund. Math., LXXXIII (1974), 269–281.
• J. M. Jędrzejewski, W. Wilczy\.nski, O rodzinie zbior\.ow ${\cal B}$-granicznych, Zeszyty Naukowe Uniwersytetu \L\.odzkiego, Seria II 52 (1973), 39–43 (in Polish).
• J. C. Oxtoby,Measure and category, Springer-Verlag (1976).
• W. Wilczyński, On the family of sets of approximate limit numbers, Fund. Math.,LXXV (1972), 169–174.