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Is there a continuous and monotone function that's nowhere differentiable ?

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up vote 8 down vote accepted

No. Even without the assumption of continuity, a monotone function on $\mathbb{R}$ is differentiable except on a set of measure $0$ (and it can have only countably many discontinuities). This is mentioned on Wikipedia, and proofs can be found in books on measure theory such as Royden or Wheeden and Zygmund. You can read the details in the latter book at this link.

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Even any function of bounded variation is differentiable almost everywhere.

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This is because a function of bounded variation is a difference of monotone functions. –  Jonas Meyer Jan 11 '11 at 8:27
    
Yes, it would have been better to start the answer with "More generally" instead of "Even". –  Shai Covo Jan 11 '11 at 12:42
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