# Inverse Function Properties

What does the inverse function say when $\det f'(x)$ doesn't equal $0$?

I know that if a function is one-to-one, than it has an inverse. However, I'm confused about the question above. Would appreciate if someone could get to me!

I'm assuming you're talking about the inverse function theorem. Basically, if you have a function $f: U \to V$ where $U$ and $V$ are open sets in Euclidean space, and furthermore, if $det(Df(p))$ where $p \in U$ is non-zero, then there is an inverse differentiable function from a neighbourhood of $f(p)$ to a neighbourhood of $p$. Furthermore, the derivative of $f^{-1}$ at $f(p)$ is $(Df(p))^{-1}$. Intuitively, this says that $f$ is $locally$ invertible (or more interestingly, a local differentiable homeomorphism).