I am new in differential/Riemannian geometry and I am working right now with the book of John Lee, Introduction of Riemannian Manifolds. He defines the distance of $p$ and $q$ ($p,q\in M$, $(M,g)$ is a Riemannian manifold) as the infimum of the length of all admissible curves, i.e. piecewise regular, such that the endpoints of the curve are $p$ and $q$. He also consider maps of the form $$c\colon \{t_o\} \to M$$ as admissible curve. (Let us say that $c(t_0)=p$.) The length of a curve $\gamma \colon [a,b] \to M$ is defined as $$L_g(\gamma):=\int_a^b |\gamma '(t)|_g\, dt.$$ What confuses me is that $c '(t_0)$ is not defined. How is the length of $c$ defined? I know that the answer should be zero, since the distance from $p$ to $p$ should be zero (if not, then it wouldn't be a distance to begin with). But is it because $$\int_a^a f(t)\, dt=0$$ for all measurable functions $f$? Or would it be better to define the length of such "point"-curves to be zero?
I know that other books, such Riemannian Geometry from Manfredo P. do Carmo, just define length and distance on piecewise smooth curves. There we can define a curve from $p$ to $p$ as $\gamma(t)=p$ for all $t\in I$ and any interval $I$. Then it is easy to see that $\gamma '(t)=0$ and the length is zero, since the integral over the zero function is zero. But I cannot adopt it to the definition above, since $\gamma$ would not be admissible anymore.