Open and surjective mapping theorems for differentiable maps with critical points

Abstract

Let $\Omega$ be an open subset of $\mathbb R^n (n \ge 2)$, and let $F : \Omega \rightarrow \mathbb R^n$ be a continuously differentiable map with countably many critical points. We show that $F$ is an open map. Let $G :\mathbb R^n \rightarrow \mathbb R^n (n \ge 1)$ be a continuously differentiable map such that $G(x) \rightarrow \infty$ as $x \rightarrow \infty$. Then it is proved that $G$ is surjective if and only if each connected component of the complement of the set of critical values of $G$ contains at least one image of $G$. Several applications of both theorems especially to complex analysis are presented.