One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces - Mykhailyuk, V. V. (Ukrains’kyi Matematychnyi Zhurnal, 2005)
We investigate the existence of a separately continuous function f : X × Y → 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 X and y_0 Y, we show that the separately continuous function f : X × Y → 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.