Suppose $X$ is a smooth vector field on a smooth manifold $M$.
For any $p \in M$, the theorems of existence and uniqueness to solutions of ODEs show that there is a unique differentiable function $\theta ^ {(p)} :J \to M$, where $J$ is an open interval containing $0$, such that $\theta ^ {(p)} (0) = p$ and $\frac{d}{dt}\theta ^ {(p)} (s)=X_{\theta^{(p)}(s)}$. Then we can define a function $\theta$ on an appropriate subset of $M \times \mathbb{R}$ by $\theta(p,s)=\theta^{(p)}(s)$. $\theta$ is the flow generated by $X$.
I am trying to understand why $\theta$ is smooth, and I don't even see why it has to be continuous. I thought of using the explicit construction from the proof of the existence theorem, but it seems to be quite involved so I wonder if there is a simpler way.
This is a consequence of a fact from ODE-theory known as "smooth dependence of solutions on intial conditions". The proof of smooth depence is substantially more difficult than the proof of local existence and uniqueness of solutions, so it will not be easy to deduce this from the proof of existence.