Any hint or demo to prove that the length of a continous parametrized curve defines a continous function in a normed space?
2026-03-25 20:36:25.1774470985
Question about the continuity of the length of a continous parametrized curve
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Define $f$ by $x\mapsto x^2\sin(1/x)$ if $x\neq0$ and $x\mapsto0$ if $x=0$. Note that $f$ is differentiable. However, its length $$\int_\Omega\sqrt{1+\left(f'\left(x\right)\right)^2}\text{d}x$$ is undefined on a closed interval $\Omega$ containing the origin and therefore not continuous.