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If a curve is Lipschitz continuous, it is rectifiable. What can be said about Hölder Continuous ones? Are they rectifiable too? Is there either proof or else an example of a function that is Holder continuous but not rectifiable?

Thank You.

azimut
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2 Answers2

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If $(B_t)_{t\geq 0}$ is a standard Brownian motion then $t \longmapsto B_t$ is almost surely Holder continuous with exponent $\alpha < \frac12$ and almost surely has infinite first variation. This gives a counterexample to Holder continuity implying rectifiability.

A proof is given here: http://www.math.nyu.edu/faculty/varadhan/fall06/fall06.1.pdf or a quick google for "brownian motion holder continuity" gives many other proofs.

I don't know of an elementary example off the top of my head though.

You may want to look at Hölder continuous but not differentiable function though. This example seems like it should have infinite first variation as well, but again I don't see an elementary proof.

Community
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nullUser
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    Just as an extra: Rectifiability can be generalized to so called bounded $p$-variation: just replace the absolute values with absolute values to the power $p$. Functions of bounded $p$-variation and Hölder continuous functions are intimately related: A function has bounded $p$-variation if and only if it can be written as a reparametrization of a $1/p$-Hölder continuous function. – Thomas Mar 13 '13 at 21:07
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The function

$$f(x) \; = \; x \sin\left(\frac{1}{x}\right)$$

with $f(0) = 0$ is Hölder continuous with Hölder exponent $\frac{1}{2}$ (but not of any higher Hölder exponent). Two proofs can be found in: Solution to Monthly Problem 3939, American Mathematical Monthly 48 #6 (June-July 1941), 413-414 (doi:10.2307/2302655). As for proving $f(x)$ is not rectificable, you can use the same method that is used in the Math StackExchange question Curve In a Closed Interval with an Infinite Length.

theHigherGeometer
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Dave L. Renfro
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