Suppose a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ is nowhere monotone. Show that there exists a local minimum for each interval. Suppose a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ is nowhere monotone. Show that there exists a local minimum for each interval.
This question is from Moscow institute. First of all, I can't even construct a nowhere monotone function. What I can think of, linear functions, clearly do not satisfy this, as we can always zoom in to get monotonicity. 
Remark: A function $f$ is nowhere monotone if for every interval $[a,b]$, $f$ is not increasing or decreasing.

Also, OP can't figure out a nowhere monotone function. I have one beautifull example >:) 
$$ \mbox{Let be }f\colon\mathbb{R}\to\mathbb{R}\mbox{ such that }f(x)=\begin{cases} x, & \mbox{ if } x\in\mathbb{Q} \\ -x, & \mbox{ if } x\notin\mathbb{Q} \end{cases} $$
such beautifull funtion, is isnt?     

