Let $f, g, L: \mathbb{R}^n \to \mathbb{R}^n$ and $L$ is a linear isomorphism, and let $|g(x)| \le M|x|^2$ on $\mathbb{R}^n$ for some $M > 0$. Prove that $f$ is locally invertible at $0$, i.e. $f$ is invertible on some open neighborhood of $0$.
I tried to use the inverse function theorem, but I failed because $f$ may be differentiable on no open neighborhood of $0$. How can I solve this?
