The following is a problem from the appendix to Chapter 13 entitled "Riemann Sums", from Spivak's Calculus
3 . Consider a curve $c$ given parametrically by two functions $u$ and $v$ on $[a,b]$. For a partition $P=\{t_0,...,t_n\}$ of $[a,b]$ we define
$$l(c,P)=\sum\limits_{i=1}^n \sqrt{[u(t_i)-u(t_{i-1})]^2+[v(t_i)-v(t_{i-1})]^2}$$
this represents the length of an inscribed polygonal curve. We define the length of $c$ to be the least upper bound of all $l(f,P)$, if it exists.
Prove that if $u'$ and $v'$ are continuous on $[a,b]$, then the length of $c$ is
$$\int\limits_a^b \sqrt{u'^2+v'^2}$$
My question is about the sentence
We define the length of $c$ to be the least upper bound of all $l(f,P)$, if it exists.
Why does the problem statement mention $l(f,P)$, ie why the function $f$ instead of $c$?
I would have expected the length of $c$ to be the least upper bound of $l(c,P)$ for all possible partitions $P$.