Given the indicator function of the rationals on $(0,1)$.
Could anyone help me understand why the measure of the set of discontinuity points is not zero?
It is pretty obvious to me that this function is not Riemann integrable. So according to Lebesgue theorem, the set of discontinuity points is not zero.
But it seems to me that the number of "jumps" in the function should be of size $|Q|$, and thus countable and with measure zero.
What am I missing here?