Function/Continuously differentiable/Derivation not 0/Local bijection/Exercise

Let ${}f\colon I\rightarrow \mathbb {R}$ be a continuously differentiable function (defined on an open interval), and let ${}a\in I$ be a point with ${}f'(a)\neq 0$. Show that there exist open intervals ${}J\subseteq I$ with ${}a\in J$ and ${}J'\subseteq \mathbb {R}$, such that the restricted function ${}f\colon J\rightarrow J'$ is bijective.

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