For the first one, without knowingwhattheorems you have available, I don't see how we can assist. I would have used "a bounded function is integrable on [a, b] if and only if its set of points of discontinuity in [a, b] has measure 0". Do you have that?

As for the second problem, consider f(x)= 1 if x is rational, -1 if x is irrational.