Points of joint continuity and large oscillations

Abstract

For topological spaces $X$ and $Y$ and a metric space $Z$, we introduce a new class $N(X × Y,Z)$ 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 $f ∈ N(X×Y,Z)$, 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 $X$.