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Let $f:A\subset \mathbb{R}^n\rightarrow \mathbb{R}^n$ an submersion of class $C^1$. Show that $f$ is a local diffeomorphism at each point in open A.

I thought so, if $f$ is a submersion then the derivative of $f$ is surjective, consequently we will not have singular points of $f$, that is, we conclude that $\det Df(p)\neq 0$ for every point $p \in A$, it follows that $Df$ is an isomorphism. Right by the Inverse Function Theorem, there are open neighborhoods $V$ and $f(V)$ with $p \in V$ and $f(p) \in f(V)$ such that $f|_{V}: V \rightarrow f(V)$ is a local diffeomorphism of class $C^1$. Is this correct?

Arctic Char
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    I think you mean $\det(Df) \neq 0$ for every point but besides that yes that's the idea! – Osama Ghani Jul 21 '20 at 17:09
  • That. The truth, the Jacobian who is different from zero. Thank you! –  Jul 21 '20 at 17:12
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    You should stress that the implication "derivative is surjective" => "$\det Df(p)\neq0$" only works because $Df(p)$ is a square matrix, since we're looking at a map from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^n$. – Thorgott Jul 21 '20 at 17:34

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