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After this question, I'd like to know more about the behavior of functions $f : \mathbb{R} \to \mathbb{R}$ where the curve $(x,f(x))$ is a rectifiable curve. Is it necessary that they are piecewise continuous ? Can they be nowhere differentiable ? or Should they be differentiable almost everywhere ?

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If it's a rectifiable curve, $f$ must be continuous, not just piecewise continuous. Moreover, $f$ must have bounded variation, so it is differentiable almost everywhere.

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The function need not be continuous. For a counter example, please see the function mentioned in this answer : math.stackexchange.com/questions/65211/… –  Rajesh D Sep 18 '11 at 5:42
    
AFAIK the usual definition of rectifiable curve requires continuity. See e.g. eom.springer.de/r/r080130.htm –  Robert Israel Sep 18 '11 at 7:13
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As already pointed out, rectifiability implies continuity.

You could instead ask the question without requiring continuity, in which case you might just ask that the graph $\{(x,f(x):x\in [a,b]\}$ has finite one-dimensional Hausdorff measure for every closed interval $[a,b]$.

However, in this case the answer is trivially negative: E.g. take the function that takes the rationals to $1$ and the irrationals to $0$. It satisfies the hypotheses, but is discontinuous, and hence nondifferentiable, everywhere.

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