Let $f \in C^1(\mathbb R^n, \mathbb R^n)$. Assume that
for all $x\in\Bbb{R}^n$ , $\det(Df(x))\neq 0$,
for all $x\in\Bbb{R}^n$ with $\|x\|=1,\|f(x)\|\geq 1$,
$f(0)=0$.
Prove that $f(U)\supseteq U$, where $U = \{x\in\Bbb{R}^n : \|x\|<1\}$.
All $\|\cdot\|$ are norms.
My teacher's hint: Prove that $f(U)\cap U$ and $f(U)^c\cap U$ are both open, and conclude that one of these sets should be empty since $U$ is connected.
I am guessing that we need to use inverse theorem which since $f(x=1)>1$ then in the neighborhood of $x=1, x<1$ and $x>1$, $f(x)$ will be less than 1 and then use the hint.