Points of joint continuity and large oscillations

For topological spaces X and Y and a metric space Z, we introduce a new class [FORMULA] 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 [FORMULA], then, for any e > 0, the projection of the set D e (f) of all points p ∈ X × Y at which the oscillation ω f (p) ≥ e onto X is a closed set nowhere dense in X.