2
$\begingroup$

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

1 Answers 1

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.

  • 0
    @August: Choose the $a_k$s such that it converges uniformly.2011-10-28