Let $F:\mathbb{R}^{2}\to \mathbb{R}^{2}$ be a smooth map where its Jacobian $$ Jf(x, y) = \det\begin{pmatrix}\frac{\partial F_{1}}{\partial x} & \frac{\partial{F_{1}}}{\partial y} \\ \frac{\partial F_{2}}{\partial x} & \frac{\partial F_{2}}{\partial y}\end{pmatrix} $$ is nonzero everywhere. Is it true that $F$ is injective? Also, if it is true, if $F$ is homeomorphism, then is it diffeomorphism?
This is true for smooth maps $F:\mathbb{R}\to \mathbb{R}$, but I believe there exists counterexamples for higher dimensional case. But I can't find it.