Suppose $V$, $W$ are banach spaces, $\alpha \in V$ and $f$ maps from an open set $U$ containing $\alpha$ to $W$.
Suppose further $f$ is continuously differentiable in $U$ (in a frechet manner), and that $df$ is a surjective map in $Hom(V,W)$, is it true that $f(U)$ is a neighborhood of $f(\alpha)$?
A claim similar to this appeared in the book Advanced Calculus by Loomis and Sternberg, where it is suggested that it follows naturally from the implicit-function theorem. However, I am having some difficulty proving this.