This is theorem 9.24 in Rudin, known as the inverse function theorem:
Suppose $f$ is a continuously differentiable map of an open set $E \subseteq \mathbb{R}^n$ into $\mathbb{R}^n$, $f'(a)$ is invertible for some $a \in E$ and $b= f(a)$. Then
(a) There exists open sets $U,V$ in $\mathbb{R}^n$ with $a \in U, b \in V$ such that $f: U \to V$ is a bijection.
(b)$f^{-1}$ is continuously differentiable.
My question: Can we add the following to the conclusion of the inverse function theorem?
(c) $f'(x)$ is invertible for all $x \in U$?
Looking at Rudin's proof, I think we can, but maybe I'm not too sure.