Points of joint continuity and large oscillations

For topological spaces X and Y and a metric space Z , we introduce a new class of mappings f : X × Y → Z containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping f from this class and any countable-type set B in Y , the set C B ( f ) of all points x from X such that f is jointly continuous at any point of the set { x } × B is residual in X : We also prove that if X is a Baire space, Y is a metrizable compact set, Z is a metric space, and , then, for any ε > 0, the projection of the set D ε ( f ) of all points p ∈ X × Y at which the oscillation ω f ( p ) ≥ ε onto X is a closed set nowhere dense in X .