An amusing sequence of functions

Abstract

We consider the amusing sequence of functions $f_n:\mathbb{R}\to \mathbb{R}$ given by $$f_n(x)=\sum _{k=1}^n\frac{|\sin (k\pi x)|}{k}.$$ Every rational point is eventually the location of a strict local minimum of $f_n$: more precisely, $f_n$ has a strict local minimum in all rational points $x=p/q\in \mathbb{Q}$ with $|q|\le \sqrt{n}$.