Let $I\subset \mathbb{R}$ be an interval, let $U\subset \mathbb{R}^n$ be open, and let

\begin{align*} F:I\times U\to \mathbb{R}^n \end{align*}

be continuous and locally Lipschitz in the state variable, locally uniformly in time. For every $(t_0,x_0)\in I\times U$, there exists an interval $J\subset I$ with $t_0\in J$ and a solution

\begin{align*} x:J\to U \end{align*}

of the initial value problem

\begin{align*} \dot{x}(t)=F(t,x(t)), \qquad x(t_0)=x_0, \end{align*}

where the differential equation is imposed at interior points of $J$ and by one-sided derivatives at endpoints of $J$ belonging to $J$, such that $x$ is maximal with respect to extension among solutions of the same initial value problem on intervals contained in $I$. Moreover, this maximal solution is unique in the following strong sense: if $y:K\to U$ is any solution of the same initial value problem on an interval $K\subset I$ with $t_0\in K$, then $K\subset J$ and $y(t)=x(t)$ for every $t\in K$.