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It is well known that there are functions $f \colon \mathbb{R} \to \mathbb{R}$ that are everywhere continuous but nowhere monotonic (i.e. the restriction of $f$ to any non-trivial interval $[a,b]$ is not monotonic), for example the Weierstrass function.

It’s easy to prove that there are no such functions if we add the condition that $f$ is continuously differentiable, so it is natural to ask the same question, with $f$ only differentiable. This seems to me a non trivial question, since, at least a priori, the derivative $f'$ could change sign on any non trivial interval, so we cannot use the standard results to prove the monotonicity of $f$.

Question: Does it exist a function $f \colon \mathbb{R} \to \mathbb{R}$ that is everywhere differentiable but nowhere monotonic?

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@Liviu: He explained that in the second line, in parentheses. –  GH from MO May 16 at 13:12
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Yes, such functions exist. Denjoy constructed an example about a century ago. There are several ways of doing it, one of the most straightforward being the one I outlined on AoPS: artofproblemsolving.com/Forum/viewtopic.php?p=244776 –  fedja May 16 at 13:18
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@fedja: Your comment should appear as an answer, and the question can be closed. –  GH from MO May 16 at 13:19
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@Todd: I did not mean that the question should be deleted. I only meant that it should be closed in the sense that there is an accepted answer for it. –  GH from MO May 16 at 13:57
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@GHfromMO Just to be clear, the notion of closure in stackexchange websites is usually understood in the sense given here: meta.stackexchange.com/questions/10582/… . (The notion of deletion is discussed here: meta.stackexchange.com/questions/5221/…) Maybe you didn't mean closure in that sense, but again to be clear for everyone, what I'm saying is that neither closure (in this sense) nor putting the question on hold would be appropriate here. –  Todd Trimble May 16 at 14:10

2 Answers 2

up vote 15 down vote accepted

Everywhere differentiable but nowhere monotonic real functions do exist. It seems that the first correct examples were found by A. Denjoy in this paper. A short existence proof, based on Baire's category theorem, was given by C. E. Weil in this paper.

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For more extreme examples, see my answer at How discontinuous can a derivative be?. Note that regarding the "levels of pathology" I list, each of #2, 4, 5, 6, 7 implies the differentiable function is nowhere monotone. (Recall that monotone in an interval implies at most countably many discontinuities in that interval.) –  Dave L Renfro May 16 at 16:56

A good reference for this is the paper

Y. Katznelson and K. Stromberg. "Everywhere differentiable, nowhere monotone, functions, Am. Math. Monthly, 81 (4), (1974), 349-353.

There, the authors give an explicit construction of a function $L:\mathbb R\to\mathbb R$ that is differentiable, with bounded derivative, and such that all three sets $\{x\mid L'(x)>0\}$, $\{x\mid L'(x)=0\}$, and $\{x\mid L'(x)<0\}$ are dense in $\mathbb R$. The argument is flexible enough that we can arrange $\{x\mid L'(x)=0\}=\mathbb Q$, if wanted. The function $L$ is the difference of two monotone functions, and $L'$ is not Riemannn integrable over any interval. (Details of the construction can be found here.)

In the paper, they indicate that the first example of a differentiable, nowhere monotone, function is due to Köpke, in 1889, with a further example due to Pereno in 1897. (I have not examined their constructions myself, so I do not know whether there are mistakes in their presentation, as the other answer suggests, but Pereno's function is discussed as an example in Hobson's book from 1957.)


For this kind of results, two obvious references to consult are Counterexamples in Analysis, by Gelbaum and Olmsted, and A second course on real functions, by van Rooij and Schikhof. The latter presents the Katznelson-Stromberg construction in detail.

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What I read suggests that Denjoy criticized Köpke's proof and the subsequent corrections to it. –  GH from MO May 16 at 13:59
    
@GHfromMO I figured. I downloaded the paper from your link (thanks!), I guess I'll be reading it rather than grading today... –  Andres Caicedo May 16 at 14:01
    
Thank you! I had checked Counterexamples in Analysis, but I didn't find the example (at least not in the list at the beginning of the book) –  Ricky May 16 at 14:06
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Here are links to the paper (jstor, so it might not be accessible to everyone): dx.doi.org/10.2307/2318996 jstor.org/stable/2318996 –  Martin Sleziak May 17 at 8:10

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