Let $X,Y$ be Banach spaces. $V\subset X$ open and $G:V\subset X \to Y$ differentiable function.
If $u\in X$ is a critical point of $G$, then the linear mapping $dG(u): X \to Y$ is not surjective.
My Questions: If $u \in X$ is not critical, then must it $dG(u)$ be surjective?
have a regular point always surjective derivative?
Is the derivative at any regular point a submersion?