# A generalization of a Baire theorem concerning barely continuous functions

## TL;DR

This paper generalizes Baire's theorem by demonstrating that functionally fragmented maps from paracompact spaces to metric spaces possess specific measurability and fragmentation properties, extending the classification of barely continuous functions.

## Contribution

It introduces a broader class of functions and establishes their measurability and fragmentation properties under general topological conditions, extending Baire's original theorem.

## Key findings

- Functionally fragmented maps are $\sigma$-discrete and $F_\sigma$-measurable.
- Such maps are Baire-one if the target space is weak adhesive.
- They are countably functionally fragmented if the domain is Lindelöf.

## Abstract

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.