Is true that $n=m$? Let $O\subseteq \mathbb{R}^m$ and $U\subseteq\mathbb{R}^n$ and $f:O\to U$ a bijective  function with $f$ and $f^{-1}$ differentiable in their domains. Is true that $n=m$?
$O$ and $U$ are open.
I don't know how to approach this problem. Any help please!!

Since $O$ and $U$ are open then $n=m$ by invariance of domain but this is a bit high powered. Notice that since both $f$ and $f^{-1}$ are differentiable then:
$f\circ f^{-1}(x)=x$ implies that $I_{n}=Df\circ Df^{-1}$
so $Df$ has a right inverse. A similar computation with $f$ and $f^{-1}$ reversed shows that $Df$ has a left inverse too. Hence, $n=m$.
