I have a proof but im not sure that it is 100% rigorous.
I started with the defnition of the length of a curve:
Integral from a to b over ||y´(t)|| dt is more or equal then
|| Integral from a to b y´(t)dt || = ||y(a)-y(b)|| and since y(t) is continuous, then both y(a) and y(b) must be finite numbers, meaning, the length of the curve is finite?
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sadcat1000
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No, it is not rigorous, because what you are showing is that the length of the curve is at least as large as a finite number. This is not sufficient to prove that the length is not infinite.
uniquesolution
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you are right i didnt notice it. Is there another way of showing it? – sadcat1000 Mar 24 '23 at 18:34
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If $f$ is differentiable, $f'$ is continuous. You can apply the extreme value theorem to $f'$. – user3257842 Mar 24 '23 at 19:11
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1@user3257842 Your comment is not true. – Mike Earnest Mar 28 '23 at 21:07
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1My mistake. $y$ however is stated to be continuously differentiable, so $y'$ is continuous. – user3257842 Mar 28 '23 at 22:41