Points of continuity of quasi-continuous functions
Fix a function \(f:X\times Y\to Z\) for topological spaces \(X\), \(Y\) and \(Z\). Then \(f\) is KC if it is quasi-continuous with respect to the first variable and continuous with respect to the second, and is Weston if \(\{x\in X\mid (x,y)\in C(f)\}\) is residual for each \(y\in Y\), where \(C(f)\) is the set of points of continuity. The space \(Y\) is Weston with respect to \(Z\) provided for all Baire \(X\) and all \(f\) if \(f\) is KC it is also Weston. Then \(Y\) is Weston with respect to every weakly developable completely regular space if and only if it is Weston with respect to \(\mathbb R\). Several conditions are given under which we can deduce that \(f\) has the Weston property, for example if \(X=\prod_{i=1}^nX_i\) where \(X_1\) is Baire, each \(X_i \ (i>1)\) has a countable \(\pi\)-base, the product of each \(X_i \ (i>1)\) with a Baire space is Baire, \(Y\) is Weston with respect to \(Z\), \(Z\) is regular and \(f\) is quasi-continuous with respect to each \(X_i\) and continuous with respect to \(Y\). Making use of a game introduced by \textit{G. Gruenhage} [General Topology Appl. 6, 339--352 (1976; Zbl 0327.54019)], conditions are given under which the set of points of continuity of a function forms a (dense) \(G_\delta\) subset of the domain.
scientific article; zbMATH DE number 7439076
scientific article