A generalization of a Baire theorem concerning barely continuous functions

We prove that if $X$ is a paracompact space, $Y$ is a metric space and $f:X\to Y$ is a functionally fragmented map, then (i) $f$ is $\sigma$-discrete and functionally $F_\sigma$-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 Lindel\"{o}ff. This result generalizes one theorem of Rene Baire on classification of barely continuous functions.