Let X: U → ℝ2 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 DXz and the global injectivity of the local diffeomorphism given by X. This set U induces a neighborhood of ∞ in the Riemann Sphere ℝ2 ∪ {∞}. In this work we prove the existence of a sufficient condition which implies that the vector field X: (U,∞) → (ℝ2, 0), -which is differentiable in U \{∞} but not necessarily continuous at ∞,- has ∞ as an attracting or a repelling singularity. This improves the main result of Gutiérrez-Sarmiento: Asterisque, 287 (2003) 89-102.