Separate and joint continuity
This paper is a survey of the most important problems concerning points of continuity of separately continuous functions. A function \(f: X\times Y\to M\) is called separately continuous if it is continuous with respect to each variable while the other is fixed. The set of points of continuity of f is denoted by C(f). In the paper three types of problems are discussed: (1) find the conditions on X and Y under which C(f) is a big (e.g. dense \(G_{\delta})\) set, (2) characterize C(f) as a subset of \(X\times Y\), (3) find a big subset A of X such that \(A\times Y\subset C(f)\). The main result of this theory seems to be the following theorem of \textit{I. Namioka} [Pac. J. Math. 51, 515-531 (1974; Zbl 0294.54010)]: if \(f: X\times Y\to M\) is separately continuous, X is regular and strongly countably complete and Y is locally compact and \(\sigma\)-compact and M is pseudo-metric, then there exists a dense \(G_{\delta}\)-set \(A\subset X\) such that \(A\times Y\subset C(f)\). The paper is well written and contains a fair amount of recent results. There are more than a hundred references and many diagrams which makes this exposition very clear. The paper will be a great help for all who are interested in this topic.
scientific article; zbMATH DE number 3979983