Some results on separate and joint continuity

Let f : X x K -> R be a separately continuous function and C a countable collection of subsets of K. Following a result of Calbrix and Troallic, there is a residual set of points x is an element of X such that f is jointly continuous at each point of {x} x Q, where Q is the Set of Y is an element of K for which the collection C includes a basis of neighborhoods in K. The particular case when the factor K is second countable was recently extended by Moors and Kenderov to any tech-complete Lindelof space K and Lindelof alpha-favorable X, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when K is a Lindelof p-space and X is conditionally sigma-alpha-favorable space. Here we add new results of this sort when the factor X is sigma(C(x))-beta-defavorable and when the assumption "base of neighborhoods" in Calbrix-Troallic's result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces. (C) 2009 Elsevier B.V. All rights reserved.