Let $M$ be a smooth manifold with boundary. Let $p\in\partial M$ and $X \in T_{p}M$. My question is: do there always exist an $\epsilon > 0$ and a smooth curve $\gamma: (-\epsilon,\epsilon) \rightarrow M$ such that $\gamma(0) = p$ and $\gamma'(0) = X$?
Note that: in the Lee's book, the author confirmed the existence for $\gamma: [0,\epsilon) \rightarrow M$ or $\gamma: (-\epsilon,0] \rightarrow M$. He also stated that it would always be possible to extend the domain to $(-\epsilon,\epsilon)$. Is it true?
Let your smooth manifold with boundary $M$ be the closed upper half-plane $\{(x,y)\in\Bbb R^2:y\ge0\}$. Take $p=(0,0)$ and the tangent vector to be $(0,1)$ (vertical). Can you really expect a curve inside $M$ to have this tangent vector at the origin?