Let $I\subset \mathbb{R}$ be an interval, let $U\subset \mathbb{R}^n$ be open, and let $F:I\times U\to \mathbb{R}^n$ be continuous and locally Lipschitz in the state variable. Let $J\subset I$ be an interval, and let $x:J\to U$ be a maximal solution of the differential equation $\dot{x}(t)=F(t,x(t))$, where the equation is imposed at interior points of $J$ and by one-sided derivatives at endpoints belonging to $J$, and maximal means that there is no solution $\widetilde{x}:\widetilde{J}\to U$ on an interval $\widetilde{J}\subset I$ with $J\subsetneq \widetilde{J}$ and $\widetilde{x}|_J=x$. Suppose that $b=\sup J$ is finite and is not the right endpoint of $I$, in the sense that there exists $s\in I$ with $b<s$. Then, for every compact set $C\subset U$ and every $a<b$, there exists $t\in J$ such that $a<t<b$ and $x(t)\notin C$.