Given that $f$ is a $C^1$-diffeomorphism from $\mathbb{R}^2$ to $f(\mathbb{R}^2)$ and that $f^{-1}$ is lipschitz from $f(\mathbb{R}^2)$ to $\mathbb{R}^2$.
Can't we say that as $f^{-1}$ is lipschitz then it is continuous , and as $\mathbb{R}^2$ is clopen then $f(\mathbb{R}^2)$ is also clopen, and since $f(\mathbb{R}^2)$ is a subset of $\mathbb{R}^2$ which is connected , then $f(\mathbb{R}^2)=\mathbb{R}^2$.
($f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ of class $C^1$)