Continuity of flows

Let $𝐹 : ℝ 𝑛 \to ℝ 𝑛$ be differentiable. Then for each $𝑥 0 \in ℝ 𝑛$ there exists $𝑇 > 0$ and a neighbourhood $𝑈$ of $𝑥 0$ such that $˙ 𝑥 = 𝐹 (𝑥)$ has continuous solutions $𝑞 (𝑡, 𝑥)$ for $𝑡 \in (- 𝑇, 𝑇)$ and $𝑥 \in 𝑈$ with $𝑞 (0, 𝑥) = 𝑥$ .¹ #m/thm/dynamics/flow