# Monotone+continuous but not differentiable

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

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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.