After an hour of googling this as you suggested, I still can't find the discussions or proof you were referring to. Can you please either send the link you found or help me with this? I don't know how to even start it.

Question

Prove that the function defined by:

f(x)={1 if x is rational, 0 if x is irrational}

is not integrable on [0,1]. Show that no matter how small the norm of the partition, ||P||, the Riemann sum can be made to have value either 0 or 1

Answer 1

Hmmm. I found a proof in about 15 seconds.

First, you need to understand what it means to be integrable: If a function is continuous on a given interval, it’s integrable on that interval.

Now, if you show that f(x) is discontinuous at any point in [0,1] then it is not integrable on the interval. This tidy proof shows that f(x) is discontinuous at every point in [0,1].

http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/L14.html