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 × Y → Z is
separately continuous, then there is a dense Gδ-set A in X such that f is
jointly continuous at each point of A × 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.
