Is flow generated by locally Lipschitz continuous vector field continuous in initial state and time

Question

Let $M$ be a manifold and $F:M\rightarrow TM$ a locally Lipschitz continuous vector field. Define escape times $$T^-, T^+: M \rightarrow [-\infty , \infty]$$ by $$T^-(x)=\inf\{t \leq 0 \mid \exists \text{ integral curve }\gamma:[t,0]\rightarrow M \text{ so } \gamma(0)=x\}$$ and $$T^-(x)=\sup\{t \geq 0 \mid \exists \text{ integral curve }\gamma:[0,t]\rightarrow M \text{ so } \gamma(0)=x\}$$ Define maximal domain for flow $$D=\{(t,x)\in\mathbb{R}\times M \mid t \in (T^-(x), T^+(x)) \}$$ At last define the flow $\Phi: D \rightarrow M$ by $\Phi(t,x) = \gamma(t)$, where $\gamma$ is the integral curve with starting point $\gamma(0)=x$. Note that since $F$ is locally Lipschitz continuous, Picard-Lindelöf's theorem assures that the integral curves are unique. Is $\Phi$ continuous and how to show it?