I've read the proof of formula for arc length . I wonder why the function has to have continuous derivative (According to the Stewart Calculus book). I mean in which part of the proof we used this assumption ? Here is the proof : https://brilliant.org/wiki/arc-length/
2026-03-25 11:01:46.1774436506
Proof of formula for the arc length
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It is assumed so that the expression $\sqrt{1 + \left( \dfrac {dy}{dx} \right)^2}$ is Riemann integrable and the approximating Riemann sums converge to the right thing.