Note on separate continuity and the Namioka property

A pair 〈B,K〉 is a Namioka pair if K is compact and for any separately continuous f:B×K→R, there is a dense A⊆B such that f is ( jointly) continuous on A×K. We give an example of a Choquet space B and separately continuous f:B×βB→R such that the restriction f|Δ to the diagonal does not have a dense set of continuity points. However, for K a compact fragmentable space we have: For any separately continuous f:T×K→R and for any Baire subspace F of T×K, the set of points of continuity of f|F:F→R is dense in F. We say that 〈B,K〉 is a weak-Namioka pair if K is compact and for any separately continuous f:B×K→R and a closed subset F projecting irreducibly onto B, the set of points of continuity of f|F is dense in F. We show that T is a Baire space if the pair 〈T,K〉 is a weak-Namioka pair for every compact K. Under (CH) there is an example of a space B such that 〈B,K〉 is a Namioka pair for every compact K but there is a countably compact C and a separately continuous f:B×C→R which has no dense set of continuity points; in fact, f does not even have the Baire property.