In Apostol's Calculus, Ch. 14 "Vector-Valued Functions" section 14.12, he derives the result that the function representing the arc length of a parametric curve is given by the integral of speed along the curve.
I have a very specific question about one single step in the derivation, the outline of which is as follows
We have a curve $\vec{r}(t)$, $t\in [a,b]$ and $\vec{r}$ is continuous on this interval.
We create a partition of $[a,b]$, $P=\{a=t_0,...,t_n=b\}$ and obtain a polygon with vertices at the points $\vec{r}(t_0),...\vec{r}(t_n)$. This polygon has length $\lvert \pi(P)\rvert =\sum\limits_{i=1}^n \lVert \vec{r}(t_i)-\vec{r}(t_{i-1}) \rVert$
If $\vert \pi(P)\rvert$ has an upper bound then it means the curve is rectifiable. It also means there is a least upper bound which is defined as the arc length between $a$ and $b$, denoted $\Lambda(a,b)$.
By definition, then, $\lvert \pi(P)\rvert \leq \Lambda(a,b) \leq M$ where $M$ is any upper bound for $\lvert \pi(P)\rvert$.
Now for my question
At this point, Apostol defines an arc length function $s$, defined as follows
$$s(t)=\Lambda(a,t)\ \text{if}\ t>a\ \ \ \ \ s(a)=0$$
He says
The statement $s(a)=0$ simply means we are assuming the motion begins when $t=a$.
Why did he have to say $s(a)=0$?
I'm not sure if $\Lambda (a,a)$ is defined. If it isn't, then defining it is an assumption, but why does this specify the start of the motion (can a motion not start at some negative $t$?)?