How to prove there is no continuous differentiable injective map from $\Bbb R^n$ to $\Bbb R$? $n\geq2$.
This is a problem in entrance to direct PHD of Tsinghua University. I got it from a webfriend. And I know a proof of the following: there is no continuous injective map from $\Bbb R^n\to \Bbb R$. Indeed, suppose $f(\Bbb R^n)$ is an interval $I$, choose the midpint $x_0$ and its inverse $v_0=f^{-1}(x_0)$, then $f: \Bbb R^n\backslash \{v_0\}\to I\backslash \{x_0\}$. Left side is connected, but right side is not. Contradiction!
But in that problem, inverse function theorem should be in used, I do not know how? since $f: \Bbb R^n\to\Bbb R$, but not $\to \Bbb R^n$.