Global asymptotic stability for differentiable vector fields of ℝ²

(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 DX_p 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}.