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The linear continuity of a function defined on a vector space means that itsrestriction on every affine line is continuous. For functions defined on$\mathbb R^m$ this notion is near to the separate continuity for which it isrequired only the continuity on the straight lines which are parallel tocoordinate axes. The classical Lebesgue theorem states that every separatelycontinuous function $f:\mathbb R^m\to\mathbb R$ is of the $(m-1)$-th Baireclass. 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 obtainthe following result. If $X$ is a Baire cosmic topological vector space, $Y$ isa Tychonoff topological space and $f:X\to Y$ is a Borel-measurable (evenBP-measurable) linearly continuous function, then $f$ is $F_\sigma$-measurable.Using this theorem we characterize the discontinuity point set of an arbitrarylinearly continuous function on $\mathbb R^m$. In the final part of the articlewe prove that any $F_\sigma$-measurable function $f:\partial U\to \mathbb R$defined on the boundary of a strictly convex open set $U\subset\mathbb R^m$ canbe extended to a linearly continuous function $\bar f:X\to \mathbb R$. Thisfact shows that in the ``descriptive sense'' the linear continuity is notbetter than the $F_\sigma$-measurability.

Linearly continuous functions and $F_\sigma$-measurability