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Let $f:U\longrightarrow \mathbb{R}^n$ a function with $U\subset \mathbb{R}^n$ open, $f$ injective of class $C^1$ (i.e. continuous with the first derivate continuous) such that $\forall x\in U$ the derivate $f´(x)=D f(x)$ is an isomorphism. Show that exists $f^{-1}:f(U)\longrightarrow U$ such that $f^{-1}\in C^1$. I've done this:

Consider the following theorem "Inverse function theorem: Let $U\subset\mathbb{R}^n$ an open subset, $a\in U$, $f:U\longrightarrow\mathbb{R}^n$ a function of class $C^1$. If $Df(a)$ is invertible, i.e. $det Df(a)\neq 0$, then there is an open set $V$ with $a\in V\in U$, and $W\subset\mathbb{R}^n$ open subset with $f(a)\in W$ such that $f\vert V$ is a difeomorphism $C^1$ from $V$ to $f(V)$."

From that theorem we can apply the inverse function theorem to each $x\in U$, so there is an open set $V_x\subset U$ with $x\in V_x$, and an open set $W_{f(x)}\subset\mathbb{R}^{n}$ with $f(x)\in W_{f(x)}$ such that $f\vert_{V_x}:V_x\longrightarrow W_{f(x)}$ is a difeomorphism.

Now, as $\bigcup_{x\in U}V_x=U$ we have that $f:=\bigcup_{x\in U} f\vert_{V_x}:\bigcup_{x\in U}V_x\longrightarrow \bigcup_{x\in U}W_{f(x)}$ is a ifeomorphism, so $f:U\longrightarrow f(U)$ is a difeomorphism, and so $f^{-1}:f(U)\longrightarrow U$.

A wish that somebody let me know if I have any mistakes, thanks!

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  • $\begingroup$ Maybe this en.wikipedia.org/wiki/Pasting_lemma will help re the issue of gluing the local functions. $\endgroup$ – user99680 Jul 11 '14 at 4:28
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    $\begingroup$ Everything is correct. It would be good if you complete the argument by giving the reason for $f^{-1}$ is of $C^1$ class. $\endgroup$ – Chellapillai Jul 11 '14 at 4:56
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You are correct. Formally , a bijective local diffeomorphism is a diffeomorphism, so your map is a local diffeomorphism as you showed ; an injection by assumption, and onto f(U) .

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