Let the ambient time interval be $I:=\mathbb{R}$, and let $F:I\times \mathbb{R}^n\to\mathbb{R}^n$ be continuous and locally Lipschitz in the state variable locally uniformly in time, meaning that for every compact interval $K\subset I$ and every compact set $C\subset\mathbb{R}^n$, there exists $L_{K,C}>0$ such that