On scatteredly continuous maps between topological spaces

Abstract

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 G δ -measurability (the preimage of each open set is of type G δ ). Also under Martin Axiom, we construct a G δ -measurable map f : X → Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V. Vinokurov.