(a) Let X: ℝ2→ℝ2 be a differentiable map (not necessarily C1) and let Spec(X) be the set of (complex) eigenvalues of the derivative DXp when p varies in ℝ2. If, for some ε>0, Spec(X)∩[0,ε)=∅ then X is injective. (b) Let X: ℝ2→ℝ2 be a differentiable vector field such that X(0)=0 and Spec(X)⊂{z∈ ℂ: R(z)<0}. Then, for all p∈ℝ2, there is a unique positive trajectory starting at p; moreover the ω-limit set of p is equal to {0}. © 2004 Elsevier Inc. All rights reserved.