Let $f:\mathbb{R}\rightarrow\mathbb{R}$ locally Lipschitz. Does the initial value problem $y'=f(y)$, $y(0)=y_0$ have a unique solution on $\mathbb{R}$ if $f\left(\frac{y_0}{2}\right)=f(2 y_0)=0$?
From Picard Lindelöf I know the existence and uniqueness of a solution on every open intervall around $0$. I don't know how to use the conditions on $f$.