Before getting started, let us go back on some definitions:
>Definition. Let $U$ be an open subset of $\mathbb{R}^n$ and let $f\colon U\rightarrow V\subseteq\mathbb{R}^n$ be a $C^1$-mapping, then $f$ is a $C^1$-diffeomorphism or globally invertible if and only if there exists $g\colon V\rightarrow U$ a $C^1$-mapping such that:
$$f\circ g=\textrm{id}_{V}\textrm{ and }g\circ f=\textrm{id}_U.$$
In other words, $f$ is a bijection whose inverse is smooth. This is not to be confused with:
>Definition. The mapping $f$ is said to be a local diffeomorphism or locally invertible if and only if when restricted to an open subset of $U$, $f$ is a diffeomorphism onto its image.
Please notice that being locally invertible around any point does not imply being globally invertible.
>Theorem. Let $U$ be an open subset of $\mathbb{R}^n$, let $x$ be a point of $U$ and let $f\colon U\rightarrow\mathbb{R}^n$ be a $C^1$-mapping. Assume that $\mathrm{d}_xf\colon\mathbb{R}^n\rightarrow\mathbb{R}^n$ is an invertible linear map, then there exists $V$ an open neighbourhood of $x$ such that $f\colon V\rightarrow f(V)$ is a $C^1$-diffeomorphism.
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Author of the notes: Antonio J. Pan-Collantes
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