It is known that all functions that are continuous and nowhere differentiable are also nowhere monotone but that there is a function that is everywhere differentiable but nowhere monotone. I have looked at one such construction but it was involved and I didn't manage to get a feel or an intuition for how it is possible to oscillate so much that you are nowhere monotone but still be well-behaved enough to be differentiable everywhere.

I guess that I am asking if anyone can explain the intuition/key idea behind those constructions.

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    $\begingroup$ Some references are mentioned here. $\endgroup$ – David Mitra Jul 31 '14 at 0:09

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