Inverse function theorem

Before getting started, let us go back on some definitions:
Definition. Let $U$ be an open subset of $R^{n}$ and let $f : U \to V \subseteq R^{n}$ be a $C^{1}$ -mapping, then $f$ is a $C^{1}$ -diffeomorphism or globally invertible if and only if there exists $g : V \to U$ a $C^{1}$ -mapping such that:

f \circ g = {id}_{V} and g \circ f = {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.

Notice that being locally invertible around any point does not imply being globally invertible.

Theorem. Let $U$ be an open subset of $R^{n}$ , let $x$ be a point of $U$ and let $f : U \to R^{n}$ be a $C^{1}$ -mapping. Assume that $d_{x} f : R^{n} \to R^{n}$ is an invertible linear map, then there exists $V$ an open neighbourhood of $x$ such that $f : V \to f (V)$ is a $C^{1}$ -diffeomorphism.