An affine connection on a smooth manifold $M$ is a map $\nabla: \mathcal{V}(M) \times \mathcal{V}(M) \to \mathcal{V}(M)$ satisfying several properties, where $\mathcal{V}(M)$ denotes the set of smooth vector fields on $M$. Now one can show that $\nabla_X Y(p)$ only depends on $X(p)$ and $Y$ on some neighborhood of $p$.
Then when defining the covariant derivative along a curve $\gamma: I \to M$ one requires that $\frac{D}{dt} V(t) = \nabla_{\dot c(t)} V$ holds for vector fields $V$ along $\gamma$ that are extendable to some neighborhood of $c(t)\subset M$.
The right hand side of the expression gets justified by the fact that the connection depends on the first argument only in the point. I don't understand, why we don't need that $\dot c(t)$ is extenable to some neighborhood of $c(t)$ as well.
A consecutive question would be: why don't we define it for rough sections of $TM$, if only points matter?