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

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.