# Linearly continuous functions and $F_\sigma$-measurability

## TL;DR

This paper investigates linearly continuous functions, proving they are of the first Baire class, characterizing their discontinuity sets, and showing $F_\sigma$-measurable functions can be extended to linearly continuous functions.

## Contribution

It establishes that linearly continuous functions are of the first Baire class and characterizes their discontinuity sets, linking linear continuity with $F_\sigma$-measurability.

## Key findings

- Linearly continuous functions on $\,\mathbb R^m$ are of the first Baire class.
- $F_\sigma$-measurable functions can be extended to linearly continuous functions.
- Discontinuity sets of linearly continuous functions are characterized.

## Abstract

The linear continuity of a function defined on a vector space means that its restriction on every affine line is continuous. For functions defined on $\mathbb R^m$ this notion is near to the separate continuity for which it is required only the continuity on the straight lines which are parallel to coordinate axes. The classical Lebesgue theorem states that every separately continuous function $f:\mathbb R^m\to\mathbb R$ is of the $(m-1)$-th Baire class. In this paper we prove that every linearly continuous function $f:\mathbb R^m\to\mathbb R$ is of the first Baire class. Moreover, we obtain the following result. If $X$ is a Baire cosmic topological vector space, $Y$ is a Tychonoff topological space and $f:X\to Y$ is a Borel-measurable (even BP-measurable) linearly continuous function, then $f$ is $F_\sigma$-measurable. Using this theorem we characterize the discontinuity point set of an arbitrary linearly continuous function on $\mathbb R^m$. In the final part of the article we prove that any $F_\sigma$-measurable function $f:\partial U\to \mathbb R$ defined on the boundary of a strictly convex open set $U\subset\mathbb R^m$ can be extended to a linearly continuous function $\bar f:X\to \mathbb R$. This fact shows that in the ``descriptive sense'' the linear continuity is not better than the $F_\sigma$-measurability.