
Continuous Functions
Let:
$\displaystyle f(x)=\left\{\begin{matrix} \frac{1}{q}, & \mbox{if }x\mbox{ is rational} \\ 0, & \mbox{if }x\mbox{ is irrational}
\end{matrix}\right.$
while:$\displaystyle q>0, x=\frac{p}{q}$ , and p,q are relatively prime.
How can I prove that f is continuous in every irrational point, and noncontinuous in rational points?
It looks as if it is not continuous at all (because between every irrational number there's a rational number, and viceverse) (Thinking)
*(Sorry for my bad English)
Can anyone please help me to find a way to solve this problem?
Thank you very much :)

This seems to explain this function in great detail:
Thomae's function  Wikipedia, the free encyclopedia