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Problem

Suppose that $f:(a,b)\to\Bbb R$ is differentiable and $f^\prime(x)=0$ on $D$, where $D$ is dense on $(a,b)$. Can we conclude that $f$ is a constant function?

Background

In calculus course, I'm told that there's a theorem stated when $D=(a,b)$. In fact, it's reducible. For example, if $D=(a,b)\backslash C$ where $C$ is at most countable, we have $f^\prime(x)=0$ for all $x\in(a,b)$, because if $k_0=f^\prime(x_0)\neq0$ for some $x_0\in C$, there's some $\xi\in C$ such that $f^\prime(\xi)=\eta$ for all $0\le\eta\le k_0$, so $C$ is uncountable.

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If you can show that $\,f(x)\,$ is a constant on $\,D\,$ then $\,f\,$ for sure is a constant in $\,(a,b)\,$...but can you? –  DonAntonio Nov 23 '12 at 13:40
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Without the requirement that $f'$ exists for all $x\in (a,b)$, the Devil's staircase would be a nice counterexample. –  Hagen von Eitzen Nov 23 '12 at 15:14
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2 Answers

up vote 2 down vote accepted

There is an everywhere differentiable non-constant function such that the set $Z=\{x: f'(x) = 0 \}$ is dense. The following paper gives an example of such a function:

Y. Katznelson and Karl Stromberg. Everywhere Differentiable, Nowhere Monotone, Functions. The American Mathematical Monthly , vol. 81, no. 4 (Apr., 1974), pp. 349-354.

According to this paper:

Examples of such functions are seldom given, or even mentioned, in books on real analysis. The first explicit construction of such a function was given by Kopcke (1889). An example due to Pereno (1897) is reproduced in [1], pp. 412-421.

...

[1] E. W. Hobson, Theory of Functions of a Real Variable II, Dover, New York, 1957

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Thanks. What about more stronger? For instance, if $C$ could be covered by a collection (at most countable) of open intervals $\{O_n\}_{n=1}^\infty$ whose total length is arbitrary small? –  Frank Science Nov 24 '12 at 3:43
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No you can't (but I don't have the example).

Look at this answer, where it is stated that the set of discontinuities of a derivative can be dense and have several other properties.

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