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Following the usual definition of length of a curve, the curve have to be rectifiable to have finite length. However for example the curve

$ \gamma \colon [0,1] \to \mathbb{R}, t \to t \cos^2(\pi/t) $

is not rectifiable. However I would say it makes perfectly sense to say it has length $1$.

So my question is, if there is a more general definition of length of a curve such that my example above has length $1$ (however such that for example the graph of the function $\gamma$ above has infinite length).

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    Why would you say that?2011-04-29
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    Since it is just a parametrization of the interval [0,1]2011-04-29
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    Check out the following rectifiable parametrizations: $\gamma \colon [0,1] \to \mathbb{R}, t \to a\cdot t$. Do they all have length $1$ ?2011-04-29
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    @Raskolnikov, They have length $|a|$, however (suppose $a > 0$) it is $\gamma([0,1]) = [0,a]$. So length $a$ makes sense.2011-04-29
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    @user3445: but in this example Raskolinikov is measuring the length of the range, whereas in yours you are measuring the length of the domain. The domain has length 1. That was the point of his question.2011-04-29
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    $\int_0^1 \sqrt{g(\dot{\gamma}, \dot{\gamma})}\mathrm{d}t$, no? So what if graph has infinite length? Curve and a graph of its parametrization are different things.2011-04-29

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