A generalization of a Baire theorem concerning barely continuous functions

Abstract

We prove that if X is a paracompact space, Y is a metric space and f:X→Y is a functionally fragmented map, then (i) f is σ-discrete and functionally Fσ-measurable; (ii) f is a Baire-one function, if Y is weak adhesive and weak locally adhesive for X; (iii) f is countably functionally fragmented, if X is Lindelöff.This result generalizes one theorem of René Baire on classification of barely continuous functions.