Let be a partition. Then the upper Darboux sum of with respect to is (why?). Thus, the infimum of all upper Darboux sums is . While the lower Darboux sum of with respect to is (why?). Thus, the supremum of all lower Darboux sums is . This means the upper integral does not agree with the lower integral. Thus, the (Dirichlet) function is not integrable on .