An amusing sequence of functions

We consider the amusing sequence of functions $f_n: $\mathbb{R}$ \rightarrow $\mathbb{R}$$givenby$$f_n(x)={\sum }_{k=1}^n\frac{\mathrm{\mid }\sin (k\pi x)\mathrm{\mid }}{k}\mathrm{.}$$Everyrationalpointiseventuallythelocationofastrictlocalminimumof$f_n$:moreprecisely,$f_n$hasastrictlocalminimuminallrationalpoints$x=p/q \in $\mathbb{Q}$$with$|q| \leq \sqrt{n}$.