An Interesting function

In A level, you usually see functions that are very nice. Have you ever thought of a function like this:
\[f(x)=\left\{\begin{matrix}1 & \mbox{if x is rational}\\ 0 & \mbox{if x is irrational}\end{matrix}\right.\]

If you want to draw a graph of the function, remember that
1. Between every pair of irrational numbers, there is a rational number.
2. Between every pair of rational numbers, there is a irrational number.

Have you thought of how to integrate such a function? Does it have an area between the graph and the x-axis?

The answer to the question reveals a very important relationship between the rational and irrational numbers.