This is a simple problem but I am confused about the results.
Suppose the $f:\mathbb{R}^2\longrightarrow\mathbb{R}^2$ is a differentiable mapping in $\mathbb{R}^2$ such that $\det(d_pf)\neq 0$ for all $p\in\mathbb{R}^2$. Has the mapping $f$ an inverse?. If it is not true, what conditions would lack for it to be?.