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This is a simple problem but I am confused about the results.

Suppose the $f:\mathbb{R}^2\longrightarrow\mathbb{R}^2$ is a differentiable mapping in $\mathbb{R}^2$ such that $\det(d_pf)\neq 0$ for all $p\in\mathbb{R}^2$. Has the mapping $f$ an inverse?. If it is not true, what conditions would lack for it to be?.

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See Global invertibility of a map $\mathbb{R}^n \to \mathbb{R}^n$ from everywhere local invertibility for a discussion of the sufficient condition $|f(x)| \to \infty$ if $|x| \to \infty$. – user48909 Nov 10 '12 at 17:11
    
I changed \mathrm{det} to \det. One of the differences is that proper spacing before and after $\det$ is automatic in expressions like $5\det A$. – Michael Hardy Nov 10 '12 at 19:09

It is false. Consider the exponential function $f:\mathbb{C}\to\mathbb{C}$, $f(z)=e^z$.

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