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

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.