Onepoint discontinuity set of separately continuous functions on the product of two compact spaces

TL;DR

This paper explores the conditions under which separately continuous functions on product spaces of compact spaces have a single point of discontinuity, linking it to the convergence of sequences of open sets.

Contribution

It characterizes the existence of separately continuous functions with a single discontinuity point on compact spaces using convergence of open sets.

Findings

Existence of such functions depends on sequences of open sets converging to the points.

Provides necessary and sufficient conditions for the discontinuity set to be a singleton.

Connects topological properties of spaces with the behavior of separately continuous functions.

Abstract

It is investigated the existence of a separately continuous function $f:X\times Y\to \mathbb R$ with an onepoint set of discontinuity for topological spaces $X$ and $Y$ which satisfy compactness type conditions. In particular, it is shown that for compact spaces $X$ and $Y$ and nonizolated points $x_0\in X$ and $y_0\in Y$ there exists a separately continuous function $f:X\times Y\to \mathbb R$ with the set $\{(x_0,y_0)\}$ of discontinuity points if and only if there exist sequences of nonempty functional open sets which converge to $x_0$ and $y_0$ in $X$ and $Y$ respectively.