Condition that implies non- zero Jacobian determinant

Question

I have trouble with the following exercise (from Fleming's functions of several variables): "Let $g: A\subseteq \mathbb{R}^n \to \mathbb{R}^n$ be of class $C^{(1)}$. Suppose that there is a number $c>0$ such that $|g(s)-g(t)|\geq c|s-t|$ for all $s,t \in \mathbb{R}^n$. Show that $Jg(t) \neq 0$ for all $t\in \mathbb{R}^n$ and $g(\mathbb{R}^n)=\mathbb{R}^n$ " I was able to show that $g$ is inyective, but I don't know if that helps to solve the problem, I also verified the case $n=1$ because I wanted to use induction, but I think it didn't help either.

Answer 1

Hint: Suppose $Jg(t_0)=0.$ Then then $\ker Dg(t_0)\ne \{0\}.$ This implies $Dg(t_0)(u)=0$ for some vector $u\ne 0.$ Consider $g(t_0+ hu) - g(t_0)$ for small real $h.$