# Proving the following function is a Diffeomorphism

We are given with $$f:R^n\to R^n$$ which is continuously differentiable and

$$||f(x)-f(y)||\ge||x-y||$$ then we have to show that it is a Diffeomorphism. What I think is that Injectivity is obvious, if we can show surjectivity and openness then Inverse function Theorem may help to prove this.

Kindly help! Thanks & regards

To apply the inverse function theorem you need to prove that the derivative $$Df(x)$$ is invertible. This follows quite easily from the definition (with the limit $$\lim_{h\to 0}\frac{f(x+h)-f(x)-Df(x)h}{\Vert h\Vert}=0$$): If $$y\neq 0$$, then take $$t$$ very small, so that $$\frac{\Vert f(x+ty)-f(x)-Df(x)ty\Vert}{\Vert ty\Vert}<\frac{1}{2}$$. This implies $$\frac{\Vert Df(x)y\Vert}{\Vert y\Vert}\geq\frac{\Vert f(x+ty)-f(x)\Vert}{\Vert ty\Vert}-\frac{\Vert f(x+ty)-f(x)-Df(x)ty\Vert}{\Vert ty\Vert}\geq1-\frac{1}{2}=\frac{1}{2},$$ so in particular $$Df(x)y\neq 0$$. Therefore $$Df(x)$$ is injective and, since it is a linear map, also surjective and hence invertible.

In particular, the Inverse Function Theorem states that $$f$$ is an open map with open image. The same proof as in https://math.stackexchange.com/a/3129225/58818 works in this case to prove that $$f$$ is surjective.