an explicit example of a function with a local strictly maximum dense set

Question

This problem looks very difficult )= Construct a continuous function, such that it set of strictly local maximum points, is the set of rationals.

score 5 · Answer 1 · answered Oct 17 '11 at 17:53

5

Let $f(x)=-\sqrt{x(1-x)}$ for $x\in[0,1]$, and let $g(x) = f(\text{the fractional part of }x)$.

Then $g(x)$ is a continuous function with very sharp local maxima at every integer.

Now, $h(x)=\sum_{k=1}^\infty a_k g(k!x)$, for some appropriate coefficients $a_k$ that make everything converge, should have the specified property.

answered Oct 17 '11 at 17:53

Would you care to elaborate a bit why $h$ would satisfy that property? – Oct 17 '11 at 19:44
1

After step $n$ of the summation, all rationals with denominator $n$ are local maxima. Furthermore, all remaining terms of the series are 0 at these points, and no term is ever positive, so one of these local maxima cannot stop being one as we to to the limit. – hmakholm left over Monica Oct 17 '11 at 20:53
(Of course one still needs to prove that the sum is continuous, and the all local maxima are rational, but I don't want to give everything away, because the original question sounded a bit like homework). – hmakholm left over Monica Oct 17 '11 at 23:05
1

How can i prove the continuity of the sum? – August Oct 28 '11 at 15:15
@August: Choose the $a_k$s such that it converges uniformly. – hmakholm left over Monica Oct 28 '11 at 15:38

1 Answers1