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Given that $F:R^n\rightarrow R^n$ is $C^1$ on $R^n$ s.t. the jacobian matrix $Df(p)$ is invertible for all $p \in R^n$, prove that f must be injective.

To show this in $R\rightarrow R$, I applied mean value theorem to show why $f$ it must be injective. However, I cannot figure out how I should approach this in $R^n$ in general.

One method I have in mind is to apply mean value theorem in $R^n$ (though I am not familiar with this theorem on $R^n$, and another is to invoke implicit function theorem. Both method seems to make sense to me, only that I don't know how to begin and articulate the proof.

What are ways to tackle this proof in $R^n$?

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This is already false in $\mathbb R^2$. Consider, for instance,$$\begin{array}{rccc}f\colon&\mathbb R^2&\longrightarrow&\mathbb R^2\\&(x,y)&\mapsto&e^x(\cos y,\sin y).\end{array}$$The Jacobian matrix is invertible at every point, but $f(0,0)=f(0,2\pi)$.

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  • $\begingroup$ Hello, could you also explain why inverse function theorem, which state that in this case I can always find some set $V$ around $x_0$, $W$ around $f(x_0)$ for all points in $R^n$ such that the function is bijective (and thus injective), fail to prove this statement for $R^n$? Thank you very much. $\endgroup$ – LHC2012 Jan 13 at 0:56
  • $\begingroup$ As my example shows, the fact that, near each point of $\mathbb R^2$, $f$ is injective, is not enough to deduce that $f$ is injective globally. $\endgroup$ – José Carlos Santos Jan 13 at 6:10

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