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Suppose that $f: [a,b] \rightarrow \mathrm{R}^n$ is continuous with a derivative f' whose norm is Riemann-integrable. To demonstrate the arclength integral formula, I'm trying to prove that, for every $n \in \mathrm N$, there exists a tagged partition $(P_n,\xi)$ of $[a,b]$ such that

|P_n| < \frac 1n\text{ and }|f(t_i) - f(t_{i-1}) - f'(\xi_i)(t_i - t_{i-1})| < \frac{t_i - t_{i-1}}n

for every interval $[t_{i-1},t_i]$ of the partition, where $\xi_i$ is the corresponding tag of the interval.

If f' is continuous, the statement is a consequence of the uniform differentiability of $f$. However, I don't think this will be useful in proving the general case. Any suggestions on how this can be proved?

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    The mean value theorem works when $n = 1$. However, only the inequality $|f(s) - f(r)| \leq M(s-r)$, where $M = \sup\limits_{t \in [r,s]} |f'(t)|$ is true when n > 1. Consider for example $f: [0,2\pi] \rightarrow \mathrm R^2$, $f(t) = (\cos{t}, \sin{t})$.2011-11-19

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