Curious function problem (EDIT: Not so curious, but didn`t see it at the time of writing)

Curious function problem (EDIT: Not so curious, but didn`t see it at the
time of writing)

This one is directly from my head and although it could be something
trivial I do not see the way to attack it but the problem looks
interesting and I want to share it with you, here it is:
Let us define function $f$ as $f(x)=\displaystyle\frac{\pi}{x}$.
Now, I wonder is there an easy way (or any way?) to prove (or disprove)
that for every interval $[x_1,x_2]\subset \mathbb R\setminus\{0\}$,
$x_1\neq x_2$ there exist irrational number $x_0\in [x_1,x_2]\setminus
\{\displaystyle\frac{a\pi}{b}|\displaystyle\frac{a}{b}\in\mathbb Q\}$,
such that $f(x_0)$ is rational number, in other words, that every interval
$[x_1,x_2]\subset \mathbb R\setminus\{0\}$ contains at least one
irrational number $x_0$ such that $f(x_0)$ is rational and $x_0$ is not
rational multiple of $\pi$.
Any ideas?