Cliquishness and Quasicontinuity of Two-Variable Maps

We study the existence of continuity points for mappings f: X x Y -> Z whose x-sections Y (sic) gamma -> f(x, y) is an element of Z are fragmentable and gamma-sections X (sic) x -> f(x, y) is an element of Z are quasicontinuous, where X is a Baire space and Z is a metric space. For the factor Y, we consider two infinite "point-picking" games G(1) (gamma) and G(2)(gamma) defined respectively for each gamma is an element of Y as follows: in the n-th inning, Player I gives a dense set D-n subset of Y, respectively, a dense open set D-n subset of Y. Then Player II picks a point gamma(n) is an element of D-n; II wins if gamma is in the closure of {gamma(n) : n is an element of N}, otherwise I wins. It is shown that (i) f is cliquish if II has a winning strategy in G(1)(gamma) for every gamma is an element of Y, and (ii) f is quasicontinuous if the x-sections of f are continuous and the set of gamma is an element of Y such that II has a winning strategy in G(2) (gamma) is dense in Y. Item (i) extends substantially a result of Debs and item (ii) indicates that the problem of Talagrand on separately continuous maps has a positive answer for a wide class of "small" compact spaces.