TITLE: A weird subset of $\mathbb R^2$
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: 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.

Added
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).