Let $X:U\to\mathbb{R}^2$ be a vector field defined on the complement of a compact set. We study the intrinsic relation between the asymptotic behavior of the real eigenvalues of the differential $DX_z$ and the global injectivity of the local diffeomorphism given by $X.$ This set $U$ induces a neighborhood of $\infty$ in the Riemann Sphere $\mathbb{R}^2\cup\{\infty\}.$ In this work we prove the existence of a sufficient condition that imply that the vector field $X:(U, \infty) \to (\mathbb{R}^2,0),$ which is differentiable in $U\setminus\{\infty\}$ but not necessarily continuous at $\infty,$ has $\infty$ as an attracting or a repelling singularity. This improves the main result of Gutierez--Sarmiento: (2003) Asterisque, {\bf 287}, 89--102.