One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces

Abstract

We investigate the existence of a separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with a one-point set of points of discontinuity in the case where the topological spaces $X$ and $Y$ satisfy conditions of compactness type. In particular, for the compact spaces $X$ and $Y$ and the nonizolated points $x_0 \in X$ and $y_0 \in Y$, we show that the separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with the set of points of discontinuity $\{(x_0, y_0)\}$ exists if and only if sequences of nonempty functionally open set exist in $X$ and $Y$ and converge to $x_0$ and $y_0$, respectively.