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Does there exist a nowhere monotonic continuous function from some open subset of $\mathbb{R}$ to $\mathbb{R}$? Some nowhere differentiable function sort of object?

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4 Answers 4

Check my answer to this question:

Is this condition sufficient to ensure monotonicity of a function?

For that function, there are enough details so you can prove the following:

For any $x \in (0,1)$ and any $\epsilon >0$ there exists a $y$ so that $0< |x-y| < \epsilon$ and $f(x)=f(y)$.

That function is continuous, nowhere differentiable and the above result shows that it cannot be monotonic at any point, since it is not locally constant.

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The Weierstrass function, mentioned in other answers, is indeed an example of a nowhere monotone function, meaning that $f$, even though continuous and bounded, is increasing at no point, decreasing at no point (and differentiable at no point as well). Details of this can be found in Example 7.16 in van Rooij, and Schikhof, A second course on real functions, Cambridge University Press, 1982.

That $f$ is increasing at $a$ means that there is a neighborhood $I$ of $a$ such that if $t<a$ is in $I$, then $f(t)\le f(a)$, and if $t>a$ is in $I$, then $f(t)\ge f(a)$. Thus, $f$ not increasing at $a$ iff any neighborhood of $a$ has points $t$ such that $(f(t)-f(a))(t-a)<0$. Being decreasing at $a$ can be stated similarly. See here.

We know that if $f$ is differentiable at $a$ and $f'(a)>0$ then $f$ is increasing at $a$, and if $f'(a)<0$, then $f$ is decreasing at $a$, so if a nowhere monotone function has a point $a$ in its domain where $f'(a)$ exists, then we must have $f'(a)=0$. It is indeed possible for a non-constant continuous increasing function $f$ to satisfy $f'(a)=0$ almost everywhere (we say that $f$ is singular). (Of course, if $f'(a)=0$ everywhere, then $f$ is constant.) The best known example of this phenomenon is Cantor's function, also known as the Devil's staircase (The link goes to O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen. The Cantor function, Expositiones Mathematicae, 24 (1), (2006), 1-37).

The above being said, note anyway that being increasing at a point is far from being increasing in a neighborhood of the point. If we require that $f$ is differentiable (and not constant), then there will be points $a$ where $f'(a)>0$ (so $f$ is increasing at $a$) or $f'(a)<0$ (so $f$ is decreasing at $a$). Nevertheless, as shown for example in Katznelson and Stromberg (Everywhere differentiable, nowhere monotone, functions, The American Mathematical Monthly, 81, (1974), 349-353) we can still find differentiable functions $f$ that are monotone on no interval. (I briefly state some properties of their example here; there used to be an accessible link to the paper, but apparently that is no longer the case.)

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The Weierstrass function is non-monotonic over any interval. I'm not sure you can prove it non-monotonic at every point.

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What do you mean by "at every point"? Monotonicity is, as far as I understand, a property over some open interval. If you mean "Has an environment such that..." or "Does not have an environment where it is monotonous" then an environment must contain an interval, since it is non-monotonic over every interval it has to be that no neighbourhood of any point is monotonic. –  Asaf Karagila May 31 '11 at 14:03
    
@Asaf: I was thinking one could say $f$ is monotonic increasing at $x_0$ if you can find $\epsilon$ such that $f(x) \lt f(x_0)$ for all $x \in (x_0-\epsilon,x_0)$ and $f(x) \gt f(x_0)$ for all $x \in (x_0,x_0+\epsilon)$. Maybe this isn't standard. –  Ross Millikan May 31 '11 at 14:39
    
So it is somewhat like a "left-minima" point... About standard terminology I can't say as I'm far far away from this field :-) –  Asaf Karagila May 31 '11 at 15:38
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@Asaf: I've seen the terminology "point of increase" and "point of decrease". And, FWIW, a Brownian motion sample path gives an example with none of either. –  Nate Eldredge May 31 '11 at 17:12
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I didn't realize this was from 2011 until I had looked up what I was going to mention. Although the participants probably know plenty about pointwise monotonicity at this point (pun intended), it would probably be useful for me to archive it here anyway. See Brown/Darji/Larsen's Nowhere monotone functions and functions of nonmonotonic type, Proceedings of the American Mathematical Society 127 #1 (January 1999), 173-182 for one literature entry point to these notions. –  Dave L. Renfro Apr 15 '13 at 17:36

Every monotonic function is almost everywhere differentiable (Theorem 4.3 - it's due to Lebesgue), so as an example of nowhere monotonic function you can just take any nowhere differentiable function (for example mentioned above the Weierstrass function).

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Oh, I haven't noticed the question was asked almost two years ago, the last edit misled me. –  Damian Sobota Apr 15 '13 at 17:31

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