Continuity points of quasi-continuous mappings

It is known that the fragmentability of a topological space X by a metric whose topology contains the topology of X is equivalent to the existence of a winning strategy for one of the players in a special two players “fragmenting game”. In this paper we show that the absence of a winning strategy for the other player is equivalent to each of the following two properties of the space X: ery quasi-continuous mapping f :Z→X, where Z is a complete metric space, there exists a point z0∈Z at which f is continuous; ery quasi-continuous mapping f :Z→X, where Z is an α-favorable space, there exists a dense subset of Z at the points of which f is continuous. t, we show that the set of points of continuity of f is of the second Baire category in every non-empty open subset of Z. Using this we derive some results concerning joint continuity of separately continuous functions.