The definition of a piecewise differentiable curve as given by Do Carmo in his Riemannian Geometry text is as follows:
A piecewise differentiable curve is a mapping $c: [a, b] \to M$ of a closed interval $[a, b] \subset R$ into $M$ satisfying the following condition:
there exists a partition $a<t_o<t_1< ... <t_{k-1}<t_k=b$ of $[a,b]$ such that the restrictions of c in each closed subset $[t_i,t_{i+1}]$, $i=0,..,k-1$, are differentiable.
The question:
Why is each subset, in which we want $c$ to be differentiable, closed? Since the derivative of a function has to do with a neighborhood of the function near the point that we are taking the derivative, how can we take the derivative at the points $a, b$ that are mentioned in the definition?
You need to take one-sided derivatives at the endpoints of each (closed) interval.
Note that the notion is standard. You can show that it is equivalent to have a differentiable extension on each closed subinterval $I_i$ to an open interval $J_i$ (of the restriction of $c$ to $I_i$).
A good reference should in fact be any more detailed book introducing line integrals or integrals in the context of complex analysis since in general we consider precisely these types of paths (often requiring $C^1$ on each subinterval, which again means $C^1$ taking one-sided derivatives at the endpoints).
Manfredo do Carmo (not Do Carmo) should also add the $C^1$ condition. How would you know that the length is defined otherwise?