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I have seen examples and proofs of functions that are everywhere continuous but nowhere monotone. However I have never seen a proof and example of a function that is everywhere continuous but monotonic at no point. Do you know of an example (preferably with a proof) or can you provide an accessible reference?

P.S Definition of monotonic at a point: Let $x$ be a real number. We say that $f$ is non-decreasing at $x$ if there is a neighborhood of $x$, $N_x$, such that $\frac{f(y)-f(x)}{y-x} \ge 0$ if $y \in N_x-\{x\}$. We say that $f$ is monotone at $x$ if $f$ is non-decreasing at $x$ or non-increasing at $x$.

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  • $\begingroup$ have you seen this? math.stackexchange.com/questions/42326/… $\endgroup$ – JC574 Jul 27 '14 at 13:45
  • $\begingroup$ @JC574 Yes, thanks. But that is about monotonicity in an interval. $\endgroup$ – Student Jul 27 '14 at 14:16
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    $\begingroup$ It seems Andres Caicedo's answer there contains what you're looking for. $\endgroup$ – David Mitra Jul 27 '14 at 14:30
  • $\begingroup$ yes I should have said, look at Andres answer. $\endgroup$ – JC574 Jul 27 '14 at 15:28
  • $\begingroup$ Thanks to everyone, I managed to get hold of the book mentioned on that answer and it has a lot of interesting stuff! $\endgroup$ – Student Jul 30 '14 at 13:25
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The Weierstraß function is indeed continuous everywhere but monotone nowhere.

See also: Nowhere monotonic continuous function

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  • $\begingroup$ but is it monotonic at no point? $\endgroup$ – Student Jul 27 '14 at 14:13
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    $\begingroup$ @Student: The Wierstrass nowhere differentiable function has, at each point, a Dini derivate equal to $+\infty$ and a Dini derivate equal to $-\infty,$ which implies that at each point the function is not monotone. Indeed, this implies the stronger result that at each point the function is not of monotonic type. The function $f$ is not of monotonic type at $x=a$ means that for each real number $m,$ the function $f(x) + mx$ is not monotone at $x=a.$ See this paper for more details. $\endgroup$ – Dave L. Renfro Jul 28 '14 at 15:43

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