A differentiable transformation of R^n at each point has an invertible derivative. Does it imply that the transformation is a global diffeomorphism?



Global invertibility
Let $f:\mathbb{R}^n\rightarrow M\subseteq\mathbb{R}^n$ be the differentiable transformation, with $M=f(\mathbb{R}^n)$. If $M\neq\mathbb{R}^n$, then obviously $f$ isn't an global diffeomorphism of $\mathbb{R}^n$. But it is global diffeomorphism between $\mathbb{R}^n$ and $M$.




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\**This was intended to be a comment not an answer, but I cannot comment, sorry. \**


