A Note on the Darboux Property of Fréchet Derivatives

Abstract

et $A$ be a subset of a Banach space $X$ and $f$ a Fréchet differentiable function on $A$ (with respect to $A$). We give a simple proof of the connectedness of the graph of $f'$ in $X\times X^*$ under relatively weak conditions on $A$. In particular, we simplify a proof by J. Malý of the connectedness of the range of $f'$ for some convex sets $A$. At the same time, we extend an older result of C. E. Weil on the connectedness of the range of $f'$ for some non-convex sets $A\subset\mathbb R^n$.