Hi,

Given this modified Dirichlet function: f(x) = 0 if x is in Q, else f(x) = x. I am wondering if this function is Darboux integrable on the interval [0, 2].

I managed to show that every lower Darboux sum is equal to zero, therefore the lower Darboux integral is 0. My intuition tells me that the upper Darboux integral is also 0, but I can't think of how to show that.

I thought of using the fact that f(x) is Darboux integrable if and only if: for every epsilon there exists a partition of [0, 2] such that $S(P) - s(P) < \epsilon$ where S, s are the lower Darboux sums. In our case $s(P)=0$ for every P so we have to show that for every epsilon there exists a partition such that $S(P) < \epsilon$, but didn't manage to come close to anything.

PS. If possible, please don't give a solution that uses Reimann's integral, I haven't studied it.