The Fr\'echet derivative is in the first Baire class

Abstract

Let $X$ be a normed linear space, $Y$ a Banach space, $G\subset X$ an open set and $f:G \to Y$ a mapping. We show that the Fr\'echet derivative $f'$ of $f$ is of the first Baire class on the (possibly empty) set $D\subset G$ where it is defined.