Prove the Jacobian Matrix (of a multi-variable function) is invertible

Question

I'm trying to solve this question: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a $C^1(\mathbb{R}^n)$ function, i.e, its partial derivatives of first order exist and continuous. for all $x,y\in \mathbb{R}^n$: $\left \| f(x)-f(y)) \right \|> \frac{1}{10}\left \| x-y \right \|$. I need to prove two things:

1. For all $x\in \mathbb{R}^n$, $det(J(f))$, i.e., the Jacobian is invertible.
2. F is surjective: $f(\mathbb{R}^n) = \mathbb{R}^n$

Thanks!