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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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1 Answers1

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The Weierstraß function is indeed continuous everywhere but monotone nowhere.

See also: Nowhere monotonic continuous function

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Sora.
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  • but is it monotonic at no point? – Student Jul 27 '14 at 14:13
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    @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. – Dave L. Renfro Jul 28 '14 at 15:43