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We show that there is a scattered compact subset K of the first Baire class a Baire space X and a separately continuous mapping f: X × K → R which is not continuous on any set of the form G × K, where G is a comeager subset of X. We also show that it is possible to have a scattered compact subset K of the first Baire class which does have the Namioka property though its function space C(K) fails to have an equivalent Frechet-differentiable norm and its weak topology fails to be σ-fragmented by the norm.

Representing trees as relatively compact subsets of the first Baire class