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.
Showing the Cantor function is not Lipschitz.
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analysis
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0Maybe you meant _isn't_ Lipschitz? – 2012-10-02
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0You are correct. I need to fix that. – 2012-10-02
1 Answers
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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.
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0I'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. – 2012-10-02
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0Precisely. The hint indicates that the slope of the secant between $x_n$ and $y_n$ is pretty large, when $n$ is large... – 2012-10-02
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1So 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 $\frac{3^n}{2^n}$. – 2012-10-02
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0Yes, equal. $ $ – 2012-10-03
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0I 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. – 2012-10-03
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0If you found $x_n$ and $y_n$, you can see that this is $=$. If you did not, I do not understand this conversation. (Note that $\leqslant$ may hold for functions as regular as one wants, for example constant.) – 2012-10-03