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 × Y → ℝ with a one-point set of discontinuity points in the case where the topological spaces X and Y satisfy conditions of compactness type. In particular, it is shown that, for compact spaces X and Y and nonisolated points x0 ∈ X and y0 ∈ Y, a separately continuous function f: X × Y → ℝ with the set of discontinuity points {(x0, y0)} exists if and only if there exist sequences of nonempty functionally open sets in X and Y that converge to x0 and y0, respectively.