Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
You can approximate the Cantor function by a sequence of piecewise linear functions whose arc lengths converge to 2, forming a lower bound on the arc length of the Cantor function. – Rahul Mar 18 '11 at 22:35
Thanks for the hint. Could you please elaborate a little bit here. How does one come up with the piecewise functions. – Shibi Vasudevan Mar 18 '11 at 23:15
up vote 11 down vote accepted

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.

share|cite|improve this answer
Yes, that is much easier than my method, and more general. – Jonas Meyer Mar 19 '11 at 1:45
Why 'nonconstant intervals must add up to at least 1"? It is not true if the 'slop' of the nonconstant part is great than 1. At least it is not 'clear' – user60933 Jan 6 '14 at 22:43

Partitions of the form $\{\frac{k}{3^n}:0\leq k\leq n\}$ would do the job of showing that the arclength is at least $2$. I don't know what you did exactly, but notice that the partition $\{0,1\}$ shows that the arclength is at least $\sqrt 2$. The partition $\{0,\frac{1}{3},\frac{2}{3},1\}$ shows that the arclength is at least $\frac{1}{3}(\sqrt{13}+1)$. And so on; but we would need some good way to keep track of things to see that the sum goes to $2$ as $n$ goes to infinity.

Edit: The previous version of my answer had a serious error, which has been fixed.

Here is an approach using the symmetry of the function. Let $A$ denote the desired arclength, and for each positive integer $k$, let $A_k$ denote the arclength of the restriction to $\left[0,\frac{1}{3^k}\right]$. I assert without proof something that is geometrically clear from the graph: $A_k=2A_{k+1}+\frac{1}{3^{k+1}}$. This comes from splitting the interval $\left[0,\frac{1}{3^k}\right]$ into thirds, and noticing that the portions of the graph on the outside thirds are congruent, while on the middle third the graph is a horizontal segment. This leads to $A$ being expressed in terms of $A_k$ as $$A=2^kA_k+\frac{1}{3}\sum_{j=0}^{k-1}\left(\frac{2}{3}\right)^j,$$ which is easily proved by induction using the prior equation. Notice that $A_k\geq \frac{1}{2^k}$ for all $k$, and $\displaystyle{\sum_{j=0}^{k-1}\left(\frac{2}{3}\right)^j}$ goes to $3$ as $k$ goes to infinity, so letting $k$ go to infinity shows that the arclength is at least $2$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.