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While studying for a calculus test I came upon the definition of arc length for the graph of a differentiable function $f(x): \mathbb{R}\to\mathbb{R}$ in an interval $[a,b]$ to be defined as $\int_a^b\sqrt{1+f'(x)^2} dx$.

This got me thinking about functions that are not differentiable in some points in $[a,b]$, in which case we can just break it down to subintervals on which the function is differentiable. But what about functions that are not differentiable anywhere like the Weierstrass function? Would it be correct to say that this function has an infinite arc length, since it contains infinitely many copies of itself?

If so, is it correct in general? That is, if $f(x)$ is a continuous function on the real numbers that is not differentiable anywhere is its arc length necessarily infinite?

Dan Asimov
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J. Doe
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2 Answers2

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According to these notes a rectifiable curve is of bounded variation, and a function of bounded variation is differentiable almost everywhere, so a nowhere differentiable function is not rectifiable.

A function must be differentiable almost everywhere in order for there to be any chance that it have finite arc length.

saulspatz
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In the book of T. Apostol "Mathematical Analysis" 2nd edition 1974 chapter 6 is devoted to the problem of rectifiable curves and bounded variation and it is very clear.

Luca Goldoni Ph.D.
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