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Lately I found there exists a function which is "continuous but nowhere monotonic".

So, now I want to know that (the title).

I'm really thank you if you give me a proof of it.

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What are you looking for? A (continuous or not) function $f\colon I\to \mathbb R$ is monotonically strictly increasing/decreasing if and only if $x>y$ implies $f(x)> f(y)$/$f(x)<f(y)$. –  Hagen von Eitzen Oct 18 '12 at 15:56
    
Well, if it's differentiable then $f'(x)\geq 0$ is sufficient. –  Alex R. Oct 18 '12 at 18:50

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