I'm having some trouble with the following question:
Let $f:\mathbb R^n\to \mathbb R^n$ be a function of class $C^1$ such that $f(0)=0$. Prove that if $1$ is not an eigenvalue of $Df(0)$, then there exists a neighbourhood $U$ of $0$ such that, $$\forall X\in U\setminus\{0\}, f(X)\neq X$$
I know that the linear map $Df(0)$ is the linear map that better approximates the behavior of the function $f$ near the point $0$. And because $Df(0)$ doesn't have 1 as eigenvalue this means that this linear map either contracts expands or rotates space, and I think that's why $\forall X\in U\setminus\{0\}, f(X)\neq X$.
The thing is that this is only my intuition and I'm not being able to formalize this on a proof, how can this be done?
Until this point, we've only learned the Banach fixed point theorem, the inverse and implicit function theorem, and all the basic results about partial derivatives, total derivatives, jacobian matrices, and gradients. How can this be done?