Separate continuity and joint continuity

Abstract

The main theorem is somewhat stronger than the following statement: Let X be either a locally compact Hausdorff space of a complete metric space, let Y be a compact Hausdorff space and let Z be a metric space. If a map f: X x Y -> Z is separately continuous, then there is a dense Gδ-set A in X such that is jointly continuous at each point of A x Y. This theorem has consequences such as Ellis’ theorem on separately continuous actions of locally compact groups on locally compact spaces and the existence of denting points on weakly compact convex subsets of locally convex metrizable linear topological spaces. © 1974 Pacific Journal of Mathematics.