Open Access
2005-2006 Generalized continuity and uniform approximation by step functions.
Christian Richter
Author Affiliations +
Real Anal. Exchange 31(1): 215-238 (2005-2006).


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.


Download Citation

Christian Richter. "Generalized continuity and uniform approximation by step functions.." Real Anal. Exchange 31 (1) 215 - 238, 2005-2006.


Published: 2005-2006
First available in Project Euclid: 5 June 2006

zbMATH: 1101.54017
MathSciNet: MR2218199

Primary: 54C08
Secondary: 41A30

Keywords: $\alpha$-continuous , $\alpha$-set , almost continuous , almost quasi-continuous , generalized continuity , nearly continuous , nearly open set , preopen set , quasi-continuous , semi-open set , semi-preopen set , somewhat continuous , step function , uniform limit

Rights: Copyright © 2005 Michigan State University Press

Vol.31 • No. 1 • 2005-2006
Back to Top