We're all familiar with the standard way to prove that the arc length of the Cantor function $C:[0,1] \rightarrow [0,1]$ is $2$. We also know that if a function $f:[a,b] \rightarrow \mathbb{R}$ is continuous and has bounded variation than the length of its graph is
$$ L = V(f_{s}, [a, b]) + \int_{a}^{b}\sqrt{1 + (f'(x))^2}\,dx, $$
where $f_{s}$ is the singular part of $f$ and $V(f_{s}, [a, b])$ is the total variation of the singular part on $[a,b]$. How can I compute the arc length of the Cantor function $C$ using the formula defined above? Mainly, how do I compute the total variation part? I've only managed to find one similar question How to compute the arclength of a singular function? but the answer is not satisfactory.