Inverse Function Theorem. Let $f: \mathbb{R}^{n} \to \mathbb{R}^{n}$ be a $C^{1}$ function. If $\det Df_{a} \neq 0$, there is open sets $U, V$ such that $f: U \to V$ is a diffeomorphism $C^{1}$ ($a \in U$ and $f(a) \in V$).

Why is the continuously differentiability necessary for this version? I'm trying to find a example with $f$ just differentiable, $\det Df_{a} \neq 0$ but $f$ is not a local diffeomorphism (at the point $a$)

Thanks for the advance!