Let X be a Baire space, Y be a compact Hausdorff space and ϕ: X → Cp(Y ) be a quasi-continuous mapping. For a proximal subset H of Y × Y we will use topological games G1(H) and G2(H) on Y × Y between two players to prove that if the first player has a winning strategy in these games, then ϕ is norm continuous on a dense Gδ subset of X. It follows that if Y is Valdivia compact, each quasi-continuous mapping from a Baire space X to Cp(Y ) is norm continuous on a dense Gδ subset of X.
We consider the question of preservation of Baire and weakly Baire category under images and preimages of certain kind of functions. It is known that Baire category is preserved under image of quasi-continuous feebly open surjections. In order to extend this result, we introduce a strictly larger class of quasi-continuous functions, i.e. the class of quasi-interior continuous functions. We show that Baire and weakly Baire categories are preserved under image of feebly open quasi-interior continuous surjections. We also give a new definition for countably fiber-completeness of a function. We prove that Baire category is preserved under inverse image of a countably fiber-complete function provided that it is feebly open and feebly continuous.