Sequential Closure of the Space of Jointly Continuous Functions in the Space of Separately Continuous Functions

Given compact spaces X and Y, we study the space S(X × Y) of separately continuous functions f : X × Y → ℝ endowed with the locally convex topology generated by the seminorms

$$ \begin{array}{lll}{\left\Vert f\right\Vert}^x={ \max}_{y\in Y}\left|f\left(x,y\right)\right|,\kern0.5em x\in X,\hfill & \mathrm{and}\hfill & {\left\Vert f\right\Vert}_y={ \max}_{x\in X}\left|f\left(x,y\right)\right|,\kern0.5em y\in Y.\hfill \end{array} $$

Under the assumption that the compact space X is metrizable, we prove that a separately continuous function f : X × Y → ℝ is the limit of a sequence (f _n ) ^∞ _n = 1 of jointly continuous function f : X × Y → ℝ in S(X × Y) provided that the set D(f) of discontinuity points of f has countable projections on X.