Let $n \in \mathbb{N}$, let $E \subset \mathbb{R}^n$ be open, let $I \subset \mathbb{R}$ be an interval with its relative topology from $\mathbb{R}$, and let $f: I \times E \to \mathbb{R}^n$ be continuous. Assume that $f$ is locally Lipschitz in the state variable locally uniformly in time: for every $(s,y) \in I \times E$ there exist a relative neighbourhood $U_s \subset I$ of $s$, an open neighbourhood $V_y \subset E$ of $y$, and a constant $L_{s,y} \geq 0$ such that $|f(t,u)-f(t,v)| \leq L_{s,y}|u-v|$ for all $t \in U_s$ and all $u,v \in V_y$. Let $(t_0,x_0) \in I \times E$, and let $x: J_x \to E$ be the maximal solution of the [initial value problem](/page/Initial%20Value%20Problem) $\dot{x}(t)=f(t,x(t))$ and $x(t_0)=x_0$, where $J_x \subset I$ is an interval containing $t_0$ and open in the relative topology of $I$. Let $J \subset J_x$ be a compact interval containing $t_0$ and contained in $\operatorname{int}_I(J_x)$, where $\operatorname{int}_I(J_x)$ denotes the interior of $J_x$ in the relative topology of $I$. If $(x_{0,k})_{k \in \mathbb{N}}$ is a sequence in $E$ such that $x_{0,k} \to x_0$ in $\mathbb{R}^n$, then there exists $k_0 \in \mathbb{N}$ such that for every $k \geq k_0$ there is a solution $x_k: J \to E$ of $\dot{x}_k(t)=f(t,x_k(t))$ and $x_k(t_0)=x_{0,k}$, and $\lim_{k \to \infty} \sup_{t \in J}|x_k(t)-x(t)|=0$.