### Uniform Approximation by Bivariate Step Functions Quasicontinuous with Respect to Single Coordinates

Christian Richter
Source: Real Anal. Exchange Volume 33, Number 2 (2007), 323-338.

#### 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.

First Page:
Primary Subjects: 54C08, 41A30
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.rae/1229619411
Mathematical Reviews number (MathSciNet): MR2458250
Zentralblatt MATH identifier: 05499472

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