# How to begin to prove that f is injective in R^n?

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 $$R^n$$?

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 $$f(0,0)=f(0,2\pi)$$.
• 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. – LHC2012 Jan 13 at 0:56
• 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. – José Carlos Santos Jan 13 at 6:10