TITLE: Arc length of the Cantor function
QUESTION [15 upvotes]: How does one find the arc length of the Cantor function? Wikipedia says that the length is 2. I can "see" that the length is atmost 2 by a simple trinagle inequality argument. I am struggling to come up with a partition P such that the arc length is atleast 2. I tried a partition of the form  {  1/ 3^n : 0 <= k <= n } but I guess I am making some mistake in my calculation so that I get the length as 3/4 instead of close to 2.

REPLY [14 votes]: Possibly the easiest way to see this is to note that any partition divides $[0,1]$ into two kinds of intervals: those on which $f$ is constant, and those on which it is not. The total length of the constant intervals can be made arbitrarily close to 1, while the total length of the nonconstant intervals in any partition is at least 1 because they add up to a displacement of 1 on the $y$-axis. The result follows.