Let $f: M \to N$ be a smooth map and $(f_*)_p: T_pM \to T_{f(p)}N$ an isomorphism. Then there exists an open neighborhood $W$ of $p$ such that $f\vert_W: W \to f(W)$ is a diffeomorphism.
This theorem is called inverse function theorem.
My question: If $f$ is bijective, can we say every open neighborhood $W$ of $p$,$f\vert_W: W \to f(W)$ is a diffeomorphism ?
If it is true, how can I prove that?