Monotone+continuous but not differentiable

Question

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

score 35 · Accepted Answer · answered Jan 11 '11 at 08:16

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.

score 15 · Answer 2 · answered Jan 11 '11 at 08:23

15

Even any function of bounded variation is differentiable almost everywhere.

answered Jan 11 '11 at 08:23

Shai Covo

24,077

22

This is because a function of bounded variation is a difference of monotone functions. – Jonas Meyer Jan 11 '11 at 08:27
16

Yes, it would have been better to start the answer with "More generally" instead of "Even". – Shai Covo Jan 11 '11 at 12:42

Monotone+continuous but not differentiable

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