Cantor's function: http://en.wikipedia.org/wiki/Cantor_function
There is an elementary way to prove that the arc length of the Cantor function is 2?
In this article (http://www.math.helsinki.fi/analysis/seminar/esitelmat/stat0312.pdf) they use the following result:
If $f:[a,b] \rightarrow \mathbb{R}$ is a continuous monotone function, then $f$ is singular if and only if $$L_a^b = |f(a)-f(b)|+|a-b|$$
But, there is a way for calculate the arc length of singular function without using this property? like using the arc length definition
If $X$ is a metric space with metric $d$, then we can define the ''length'' of a curve $\!\,\gamma : [a, b] \rightarrow X$ by $$\text{length} (\gamma)=\sup \left\{ \sum_{i=1}^n d(\gamma(t_i),\gamma(t_{i-1})) : n \in \mathbb{N} \text{ and } a = t_0 < t_1 < \cdots < t_n = b \right\}. $$
where the sup is over all $n$ and all partitions $t_0 < t_1 < \cdots < t_n$ of $[a, b]$.