Theorem: Suppose ${\bf g} :\mathbb{R}^n \mapsto \mathbb{R}^n$ is continuously differentiable, there exists a set $\mathcal{A} \subset \mathbb{R}^n$ such that $\bf g$ is uniformly Lipschitz on $\mathcal{A}$, and for all points ${\bf x}_0\in\mathcal{A}$ and $t>0$ there exists a unique solution ${\bf x} (t) \in \mathcal{A}$ to the ODE $\dot{{\bf x}}(t)= {\bf g} ( {\bf x} (t))$ such that ${\bf x} (t) = {\bf x}_0$, then the set of initial points that converge to an unstable equilibrium point has measure zero.
I need the followingaforementioned result for some work that I am doing. I have been unable to find any similar result in the the continuous case. I have a proof, but would like to try a bit harder to see if someone has done it already before making part of the manuscript when I could reference it. The core of the proof is the center-stable manifold theorem
Theorem: Suppose $\bf{g}:\mathbb{R}^n\mapsto\mathbb{R}^n$ is continuously differentiable, there exists a set $\mathcal{A}\subset\mathbb{R}^n$ such that $\bf{g}$ is uniformly Lipschitz on $\mathcal{A}$, and for all points $x_0\in\mathcal{A}$ and $t>0$ there exists a unique solution $x(t)\in\mathcal{A}$ to the ODE $\dot{x}(t)=g(x(t))$ such that $x(t)=x_0$, then the set of initial points that converge to an unstable equilibrium point has measure zero.w
