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2007/2008 Uniform Approximation by Bivariate Step Functions Quasicontinuous with Respect to Single Coordinates
Christian Richter
Real Anal. Exchange 33(2): 323-338 (2007/2008).

Abstract

Quasicontinuity with respect to one coordinate and symmetrical quasicontinuity strengthen the concept of classical quasicontinuity of a bivariate function $f$ from a product space $X \times Y$ into a topological space~$Z$. For certain spaces $X,Y$, we show that a function $f$ from $X \times Y$ into a metric space $Z$ is quasicontinuous with respect to the first coordinate if and only if it is the uniform limit of step functions quasicontinuous with respect to the first coordinate. This applies in particular to arbitrary $X \subseteq {\mathbb R}^m$, $m \ge 0$, and every $Y \subseteq {\mathbb R}^n$, $n \ge 1$, without isolated points. A second result concerns spaces $X,Y$ such that every continuous $f:X \times Y \rightarrow Z$ is the uniform limit of symmetrically quasicontinuous step functions. It comprises all $X,Y \subseteq {\mathbb R}$ without isolated points.

Citation

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Christian Richter. "Uniform Approximation by Bivariate Step Functions Quasicontinuous with Respect to Single Coordinates." Real Anal. Exchange 33 (2) 323 - 338, 2007/2008.

Information

Published: 2007/2008
First available in Project Euclid: 18 December 2008

MathSciNet: MR2458250

Subjects:
Primary: 41A30 , 54C08

Keywords: bivariate function , quasicontinuous with respect to one coordinate , step function , symmetrically quasicontinuous , uniform limit

Rights: Copyright © 2007 Michigan State University Press

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Vol.33 • No. 2 • 2007/2008
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