I am trying to prove: If $F$ is $C^1$ $:U\subset\Bbb R^n\to\Bbb R^n$ with $DF(0)=I$, then exists $r>0$ such that $DF(x)$ is invertible for all $x \in B_r(0)$.
I think this should be natural. Since $F$ is $C^1$, $\forall \epsilon>0, \exists r>0$ such that $\forall y\in B_r(0), ||DF(x)-DF(0)||_2<\epsilon$ and their determinant should be close. But it can go complicated in the computation of determinant. Is there any other way to prove this? (or this is wrong?)