How to interpret integration of a discontinous function My teacher says that integration of a function is the area under the curve.
But what about integration of a discontinuous function.
How do we visualize or interpret the integration of a discontinuous function like Thomae's function.

General discontinuous functions are not integrable (consider for example a function that has a discontinuity over an interval [a,b], such that a < b), but in this case, the discontinuities are the set of all rationals. In order to integrate the thomae's function, you have to integrate the latter term (namely, x=0) over a bounded interval.
Another example: consider such a function that it equals 0 in every point that is divisible by some number and only that number, and at every other point it is the signum function of the argument. The definite integral of the function (correct me if I'm wrong) is 0 in every that is divisible by some number, and |x| in every other point.
