Separate continuity, joint continuity, the Lindelِf property and p-spaces

In this paper we prove a theorem more general than the following. Suppose that X is Čech-complete and Y is a closed subset of a product of a separable metric space with a compact Hausdorff space. Then for each separately continuous function f : X × Y → R there exists a residual set R in X such that f is jointly continuous at each point of R × Y . This confirms the suspicions of S. Mercourakis and S. Negrepontis from 1991.