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Is it possible for function $f : \mathbb{R} \to \mathbb{R}$ have a maximum at every point in a countable dense subset of its domain ? The motivation for this question is I have a sequence of functions $\{f_n\}$ where the number of maxima increases with $n$ and I am interested to know what happens to the sequence of functions.

PS : every function of the sequence has a finite number of maxima.

EDIT : $f$ should not be constant function.

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    What if the function $f$ was constant?... ;) Or do you want to have the set of maximas to be countable? If you can allow ''more'' maximas, the constant function would be an example of one such function.2011-06-03
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    @Patrick: Or, if you insist on countable, the characteristic function of the rationals.2011-06-03
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    @Patrick : maxima means second derivative should be negative.2011-06-03
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    @Rajesh: So you want your function to be twice differentiable. Please include that in the statement of your question.2011-06-03
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    @Yuval : Ok, i do not need it to be differentiable but i do not want constant function either.2011-06-03
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    I was just about to write the example that Jonas Meyer suggested. Do you need anything to be satisfied by $f$ or you just wanted to know if some whatever-function $f$ would do the trick?2011-06-03
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    @Patrick : thank you for the comment about constant function.2011-06-03
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    The word *maxima* is the **plural** of *maximum*. Maxima = maximums.2014-05-13

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Thomae's function has a strict local maximum at each rational number.

I believe the Weierstrass function is another example.


Another question on this site posed the problem of showing that if $f$ is continuous and not monotone on any interval, then $f$ has a local maximum at each point in a dense subset of $\mathbb{R}$.

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    You're quick, aren't you. XD2011-06-03
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    I'd like to know if it's possible for the sequence of functions $\{f_n\}$ where each one is smooth to converge to such a function ?2011-06-03
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    @Rajesh: The Weierstrass function is a uniform limit of smooth functions.2011-06-03
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Sample paths of Brownian motion have this property (with probability $1$), see here.

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    See the third item in that subsection (Qualitative properties).2011-06-03