I missed two weeks' worth of classes in my Real Analysis II course die to personal issues, and while going over past exam questions for midterm revision, I came across some problems that I had trouble even attempting to try, due to lack of background knowledge.
Let $L$ be a vector field from $\mathbb{R}^n$ to $\mathbb{R}^n$. Let $f(x)=L(x)+g(x)$ with $L$ being a linear isomorphism and $g$ of class $C^1$ satisfying $||g(x)|| \leq M||x||^2$ for some fixed positive $M$. Is $f$ locally invertible near $0$? (that is, does some open neighborhood $U$ of $0$ exist, with $f$ restricted to $U$ being invertible from $f(U)$)
Thanks in advance for any help given, may it be hints, guidance on which material to look at, a rough sketch of a solution, et cetera.