So I know that the characteristic function of the rationals is not Riemann integrable and we can show this by showing that the upper and lower sums are different but I have a theorem in my notes which states (we have not proved) that a function, $f$, is Riemann integrable iff the set of all points where $f$ is not continuous is a set of zero measure.
So I was wondering if the following reasoning is ok:
We know that $[0,1]$ has meassure 1 and the set $\mathbb{Q}$ is a set of zero measure so it must be the case that $[0,1] - \mathbb{Q}_{[0,1]}$ is of measure $1$, so $f$ is not Riemann integrable.
(I know that this question is not very well informed, but I have this theorem and so I can use this in my exam as a quick way to show a function is not Riemann integrable, I would be more than happy for more information though)
Thanks very much for any help
