be continuous. Assume that $F$ is locally Lipschitz in the state variable locally uniformly in $(t,p)$: for every $(\tau,z,\pi)\in I\times U\times P$, there exist neighbourhoods $I_\tau\subset I$ of $\tau$, $U_z\subset U$ of $z$, $P_\pi\subset P$ of $\pi$, and a constant $L_{\tau,z,\pi}\geq 0$ such that
where $\operatorname{dist}(C,\varnothing)=+\infty$ by convention. Then there exist neighbourhoods $N\subset U$ of $x_0$ and $Q\subset P$ of $p_0$ such that, for every $(y_0,p)\in N\times Q$, the initial value problem