Quasi-continuité, continuité séparées, et topologie extrémale

Let $E$ be a Banach space, ${X_1}, \ldots ,{X_n}$ strongly countably complete regular spaces and $\Phi :{X_1} \times \cdots \times {X_n} \to E$ a function. We prove, particularly by means of techniques borrowed from J. P. R. Christensen, that if $\Phi$ is separately continuous and quasi-continuous when $E$ is equipped with the extremal topology, then $\Phi$ is jointly continuous at each point of a dense ${G_\delta }$ subset of ${X_1} \times \cdots \times {X_n}$ when $E$ is equipped with the norm topology. Various properties--new or already known--are obtained by using this result.