Real Analysis Exchange

Generalized continuity and uniform approximation by step functions.

Christian Richter

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Given two topological spaces $X$ and $Y$ and a family ${\mathcal O}_\ast$ of subsets of $X$, a function $f: X \rightarrow Y$ is called ${\mathcal O}_\ast$-continuous if $f^{-1}(V) \in {\mathcal O}_\ast$ for every open set $V \subseteq Y$. An ${\mathcal O}_\ast$-step function is meant to be a function $\varphi: X \rightarrow Y$ that is piecewise constant on a partition of $X$ into sets from ${\mathcal O}_\ast$. Using some technical assumptions on $X$, $Y$, and ${\mathcal O}_\ast$ we give representations of ${\mathcal O}_\ast$-continuous functions as uniform limits of ${\mathcal O}_\ast$-step functions. We deal in particular with $\alpha$-continuous, nearly continuous, almost quasi-continuous, and somewhat continuous functions. The paper is motivated by a corresponding characterization of quasi-continuous functions.

Article information

Real Anal. Exchange, Volume 31, Number 1 (2005), 215-238.

First available in Project Euclid: 5 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54C08: Weak and generalized continuity
Secondary: 41A30: Approximation by other special function classes

Generalized continuity $\alpha$-continuous quasi-continuous nearly continuous almost continuous almost quasi-continuous somewhat continuous $\alpha$-set semi-open set nearly open set preopen set semi-preopen set step function uniform limit


Richter, Christian. Generalized continuity and uniform approximation by step functions. Real Anal. Exchange 31 (2005), no. 1, 215--238.

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