Arc length of nowhere differentiable arcs.

Question

Does there exist an arc (something homeomorphic to the interval $[0, 1]$) that is nowhere differentiable? If so, how does one define the arc length function along such an arc? Can a topological arc have infinite length?

Answer 1

You're looking for the notion of a rectifiable curve. It's possible to talk about the length of a general curve by approximating it with line segments. As those line segments in the approximation get smaller, their total length will increase. If it doesn't go to infinity, then it approaches a value, defined to be the arc length of the curve, and the curve is said to be rectifiable. You can show that if the curve is differentiable, this process will give the same result for the arc length as the more familiar $$\int \sqrt{(x')^2+(y')^2}\,dt.$$ (See https://en.wikipedia.org/wiki/Arc_length for more details.)

As to the question of whether a nowhere differentiable curve can be rectifiable or not, using this definition one can show that a nowhere differentiable curve is never rectifiable (see e.g. this question).