following is the inverse function theorem:
Let $f: X \rightarrow Y$ is a function which is continusoly differentiable at $a \in X$ and $f(a) \neq 0$. Then $\exists f'^{-1}(a)$ and $f^{-1}$ is differentiable at $x=a$ and $f^{-1}(f(a)) = {1\over f'(a)}$
Question: How could one prove that "continuity and differentibility guarantees the existence of inverse mapping"?