Let $U$ be an open set in $\mathbb{R^n}$, and let $f: U \to \mathbb{R^n}$ be a $C^1$ function such that, for all $x$ $\in$ $U$, $Df(x)$ is an isomorphism.
My goal is to show if $f$ is one-to-one, then $f$ has an $C^1$ inverse function. I also showed that $f(U)$ is an open set in $\mathbb{R^n}$ but I can't find a way to verify that $f^{-1}$ will exist and be $C^1$ function. Any suggestions?