Because any upper Darboux sum is 1 while any lower Darboux sum is 0.
This means the upper integral is 1 while the lower integral is 0.
Thus the function is not Riemann integrable.
i want to prove that the function f:[0,1] -> R where [0,1] is the closed interval 0,1 and R is the set of reals, defined by f(x)=1 if x is rational and 0 otherwise, is not riemann integrable. i know that this is the dirischlet's function and that it is not integrable, but i'm not sure how to go about proving it. thanks for any help!