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Is there a path-connected subset of $\mathbb R^2$ such that any path connecting 2 distinct points in that subset has infinite length? I am told that there is such a set, but I don't know what it is. Thank you.

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Something like a Koch snowflake should work. – Chris Eagle Feb 13 '12 at 20:33
There is, for example almost every path of a standard Brownian motion. – Did Feb 13 '12 at 20:34
Would $\{x \}$ count? – Matt Feb 13 '12 at 20:36
@Matt I don't believe that's in the spirit of the question. – Austin Mohr Feb 13 '12 at 20:46
up vote 4 down vote accepted

The graph of any continuous nowhere differentiable function $\mathbb R\to\mathbb R$ is an example (or any continuous function that is not of bounded variation on any subinterval).

If $f:[a,b]\to\mathbb R$ is of bounded variation, then $f'$ exists in $(a,b)$ except on a set of measure $0$. Thus if $f:\mathbb R\to\mathbb R$ is nowhere differentiable, then $f$ is not of bounded variation on any subinterval.

Suppose that $f:\mathbb R\to\mathbb R$ is continuous and nowhere differentiable. Then $G=\{(x,f(x)):x\in\mathbb R\}\subset \mathbb R^2$ is path connected, and if $a<b$, then the length of any path in $G$ connecting $(a,f(a))$ to $(b,f(b))$ is bounded below by the total variation of $f$ on $[a,b]$, hence is infinite.


Exercise 26 in Chapter 3 on page 49 of Wheeden and Zygmund's Measure and integral gives a hint of a way to construct a continuous function on $[0,1]$ that is not of bounded variation on any subinterval, with no reference to differentiation, using a modification of the construction of the Cantor function.

I just looked up the examples Gerry referred to, and the third example there is essentially the same as the original example I gave yesterday (after his answer). But the exercise from Wheeden and Zygmund gives a way to construct an example somewhat different from the four mentioned in Gelbaum and Olmsted, with the possible exception of the example in Carathéodory's book (I don't know what it is).

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Gelbaum and Olmsted, Counterexamples in Analysis, Chapter 10, #18: "A simple arc that is of infinite length between every pair of distinct points on the arc" (page 141) gives four examples. Unfortunately, each one refers to previous examples (or, in one case, to page 190 in a German book by Caratheodory from 1927), so it's too long to write out any of the examples here. But the student who is interested in weird examples will not find a better resource than Gelbaum and Olmsted.

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