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?

Question

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.

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. — Patrick Da Silva, Jun 03 '11 at 05:04
@Patrick: Or, if you insist on countable, the characteristic function of the rationals. — Ross Millikan, Jun 03 '11 at 05:06
@Patrick : maxima means second derivative should be negative. — Rajesh D, Jun 03 '11 at 05:07
@Rajesh: So you want your function to be twice differentiable. Please include that in the statement of your question. — Yuval Filmus, Jun 03 '11 at 05:08
@Yuval : Ok, i do not need it to be differentiable but i do not want constant function either. — Rajesh D, Jun 03 '11 at 05:10
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? — Patrick Da Silva, Jun 03 '11 at 05:11
@Patrick : thank you for the comment about constant function. — Rajesh D, Jun 03 '11 at 05:37
The word maxima is the plural of maximum. Maxima = maximums. — bof, May 13 '14 at 03:19

score 4 · Accepted Answer · edited Apr 13 '17 at 12:21

4

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 $\mathbb{R}$.

edited Apr 13 '17 at 12:21

Community

1

answered Jun 03 '11 at 05:09

Jonas Meyer

53,602

You're quick, aren't you. XD – Patrick Da Silva Jun 03 '11 at 05:13
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 ? – Rajesh D Jun 03 '11 at 05:20
1

@Rajesh: The Weierstrass function is a uniform limit of smooth functions. – Jonas Meyer Jun 03 '11 at 05:22

score 3 · Answer 2 · answered Jun 03 '11 at 05:15

3

Sample paths of Brownian motion have this property (with probability $1$), see here.

answered Jun 03 '11 at 05:15

Shai Covo

24,077

See the third item in that subsection (Qualitative properties). – Shai Covo Jun 03 '11 at 05:17

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?

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