Prove that $\Vert f(x) -f(y)\Vert\geq (1-k)\Vert x-y\Vert,\;\text{and}\;\Vert f'(x)h\Vert\geq (1-k)\Vert h\Vert,\;\forall\,x,y,h\in\Bbb{R^n}$

Question

Good day all! I'm preparing for a Graduate exam, so I need to solve this problem.

Let $f:\Bbb{R}^n\to\Bbb{R}^n$ be a function of class $C^{1}$. We suppose that there exists $k\in ]0,1[$ such that $$\Vert g'(x)\Vert\leq k,\forall\;x\in\Bbb{R}^n$$ where $f(x)=g(x)+x,\forall\;x\in\Bbb{R}^n$

1. Prove that $\Vert f(x) -f(y)\Vert\geq (1-k)\Vert x-y\Vert,\;\forall\,x,y\in\Bbb{R^n}$

2. Prove that $\Vert f'(x)h\Vert\geq (1-k)\Vert h\Vert,\;\forall\,x,h\in\Bbb{R^n}$

3. Prove that $f$ is $C^1-$diffeomorphism from $\Bbb{R}^n\to\Bbb{R}^n$.

I started this: $$\Vert g'(x)\Vert\leq k,\forall\;x\in\Bbb{R}^n,\;\text{for some}\;k$$ $$\Vert f'(x)-1\Vert\leq k,\forall\;x\in\Bbb{R}^n,\;\text{for some}\;k$$ $$\Vert \lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}-1\Vert\leq k,\forall\;x\in\Bbb{R}^n,\;\text{for some}\;k$$

But I couldn't proceed. Please, I need an instant help. Proofs and references are welcome!

Answer 1

(Just an hint for 1, to show how to proceed.)

Your assumption on $g$ says that $g$ is $k$-Lipschitz, i.e. $$ \|g(x) - g(y)\| \leq k \|x-y\| \qquad\forall x,y\in\mathbb{R}^n. $$ Hence, $$ \|f(x) - f(y)\| = \|g(x) + x - g(y) - y\| \geq \|x-y\| - \|g(x)-g(y)\| \geq (1-k)\|x-y\|. $$