I'm trying to prove the following statement:
Let $f:\mathbb{R}\to\mathbb{R}$ a locally Lipschitz function such that $f(x_1)=f(x_2)=0$ with $x_1<x_2$. If $x_0\in(x_1,x_2)$, then the solution of the IVP $$x'=f(x),\;x(0)=x_0$$ is defined for all $t\in\mathbb{R}$.
If one look at the phase plane it makes sense, but I don't know how to start the proof, I think it involves Picard–Lindelöf theorem. Any hint?