Definition of the tangent space for a submanifold with boundary in terms of regular curves

Question

Let $d\in\mathbb N$, $M\subseteq\mathbb R^d$ and $x\in M$. $\gamma$ is called curve on $M$ through $x$ if $\gamma:I\to M$ for some nontrivial interval $I\subseteq\mathbb R$ with $0\in I$ and $\gamma(0)=x$. In that case, $\gamma$ is called regular if $\left.\gamma\right|_{I^\circ}\in C^1(I^\circ)$ and $\gamma'$ has a continuous extension to $I$.

We know define the tangent space of $M$ at $x$ to be $$T_x\:M:=\left\{\gamma'(0):\gamma\text{ is a regular curve on }M\text{ through }x\right\}.$$

Am I correct that the reason why we do not simply define a curve to be defined on an open interval is that this definition would not yield an appropriate definition of $T_x\:M$ if $M$ is a submanifold with boundary? In any case, why is it guaranteed that the extension of $\gamma'$ is unique?

Answer 1

1. Yes. The image of such an open interval w.r.t that curve would not be (relatively) open.
2. It is not. And it doesn't have to be. You only need that all its extensions have the same value $\gamma'(0)$ when extended. Since tangent vectors are an equivalence class of curves at a certain point $p$ having the same derivative (w.r.t. any chart) all these extensions will be equivalent.

Maybe one quick note. If $p$ is a boundary point, then the set of all tangent vectors is just a half space, since there is a missing outward direction. This can be completed by saying, that $T_pM$ is the smallest vector space containing all tangent vectors mentioned above.