Showing the Cantor function is not Lipschitz.

Question

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.

Did · Accepted Answer · 2012-10-02 17:50:24Z

up vote 3 down vote accepted

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.

answered Oct 2 '12 at 17:50

Did
102k1073179

	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. – abet Oct 2 '12 at 19:26
	Precisely. The hint indicates that the slope of the secant between $x_n$ and $y_n$ is pretty large, when $n$ is large... – Did Oct 2 '12 at 19:29
	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 $\frac{3^n}{2^n}$. – abet Oct 2 '12 at 20:26
	Yes, equal. $ $ – Did Oct 3 '12 at 4:58
	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. – abet Oct 3 '12 at 6:52

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