On scatteredly continuous maps between topological spaces

A map f:X→Y between topological spaces is defined to be scatteredly continuous if for each subspace A⊂X the restriction f|A has a point of continuity. We show that for a function f:X→Y from a perfectly paracompact hereditarily Baire Preiss–Simon space X into a regular space Y the scattered continuity of f is equivalent to (i) the weak discontinuity (for each subset A⊂X the set D(f|A) of discontinuity points of f|A is nowhere dense in A), (ii) the piecewise continuity (X can be written as a countable union of closed subsets on which f is continuous), (iii) the Gδ-measurability (the preimage of each open set is of type Gδ). Also under Martin Axiom, we construct a Gδ-measurable map f:X→Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V. Vinokurov.