This is one I am having a lot of difficulty with. I'm not sure how to show that the Cantor function (or 'Devil's Staircase) is not Lipschitz.

  • $\begingroup$ Maybe you meant isn't Lipschitz? $\endgroup$ – k.stm Oct 2 '12 at 17:44
  • $\begingroup$ You are correct. I need to fix that. $\endgroup$ – emka Oct 2 '12 at 17:48

Hint: For every nonnegative integer $n$, find some points $x_n$ and $y_n$ such that $|x_n-y_n|=1/3^n$ and $|f(x_n)-f(y_n)|=1/2^n$. Conclude.

  • $\begingroup$ I'm not sure I follow. I guess I may be misunderstanding what Lipschitz means. In my mind I like to think of a function having the Lipschitz property as saying that the secants are bounded by a positive M. $\endgroup$ – emka Oct 2 '12 at 19:26
  • $\begingroup$ Precisely. The hint indicates that the slope of the secant between $x_n$ and $y_n$ is pretty large, when $n$ is large... $\endgroup$ – Did Oct 2 '12 at 19:29
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    $\begingroup$ So would this be where you are going: $$\frac{|f(x_n)-f(y_n)|}{|x_n-x_y|} \leq \frac{3^n}{2^n}$$. I guess I'm still not sure what there is that $\leq$. In my mind, isn't is exactly equal to $\frac{3^n}{2^n}$. $\endgroup$ – emka Oct 2 '12 at 20:26
  • $\begingroup$ Yes, equal. $ $ $\endgroup$ – Did Oct 3 '12 at 4:58
  • $\begingroup$ I noticed that my previous comment made marginal sense. What I meant to say is: why is it $\leq$ and not $=$? My gut feeling is to claim this $\leq$, but I don't have a defense for it. $\endgroup$ – emka Oct 3 '12 at 6:52

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