An Interesting Problem

A lot of people don't really know what Analysis is all about. I was in this group for a long time, and feared it due to my lack of interest in computation, which is what usually followed in most uses of calculus and differential equations. However, I'm rather enjoying some of the work I'm doing in my Elementary Analysis course, and I look forward to getting some more challenging problems. Here's one that I found to be quite interesting:

For each rational number x, write as p/q, where p, q are integers with no common factors and q > 0, and then define the function f(x) = 1/q. Also define f(x) = 0 for all irrational real numbers. Show that f is continuous over the irrationals, but discontinuous on the rationals.

The irrational part was left out of my required homework, but I'm an overachiever. Look forward to the solution on Thursday.