Let
$X:U\to\mathbb{R}^2$ be a differentiable 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
which implies 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 Guti\'{e}rrez--Sarmiento: Asterisque, {\bf 287} (2003) 89--102.