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Let $f$ be a function defined on $[0,1]$ by

$$f(x) = { 0, \text{ if } x = 0} $$ $$f(x) = { x \sin \frac 1 x , \text{ if } 0 < x \leq 1} $$

Prove that the curve $\{(x, f(x)) : x \in [0,1]\}$ is not rectifiable.

I'm not sure how to approach this. The general idea seems logical, we're proving that the length of the curve is infinite, but the method seems difficult to find.

  • 1
    You know how to form the arclength integral, no?2011-12-12
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    it's the integral of the absolute value of the derivative of any parametrization. i'm not sure where to go from there.2011-12-12
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    Form the arclength integral and see if you can evaluate it.2011-12-12

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