$\not\exists$ bijective function $f:\Bbb{R}^n\to A(\subset\Bbb{R}^m~m>n)$such that $f$ and $f^{-1}$ are both differentiable.

Question

Suppose $m>n$. Let $A\subset \Bbb{R}^m$ such that $f:\Bbb{R}^n\to A$ is bijective. Show that there exists no such function $f$ such that $f$ and $f^{-1}$ are both differentiable.

If $m=n$ then inverse function theorem implies the above statement is wrong. But I can't fathom the relationship between $Df(x)$ and $Df^{-1}(x)$ if they exist, in the case $m\ne n$.