A solution of a problem by M. Henriksen and R.G. Woods

Henriksen and Woods [Topology Appl. 97 (1999) 175–205, Problem (C), p. 203] asked whether there are Tychonoff spaces X and Y with X×Y being Baire such that: separately continuous function f :X×Y→R has a dense (in fact: Gδ) set C(f) of points of continuity; exists a separately continuous function g :X×Y→R for which C(g) fails to contain either A×Y or X×B for any dense Gδ set A⊂X or dense Gδ set B⊂Y. l answer this question by showing the spaces X and Y can even be complete metric and condition (b) can be strengthened to the following: There exists a separately continuous function g :X×Y→R so that if C(g) contains either A×Y or X×B, then both A and B are empty.