>>11508343A function is integrable if the supremum of the lower darboux sums approaches the upper bound and vice versa for the upper sums, then yeas the integration is convergent. Simply being bounded is not enough for the integration to uniformly converge however, consider a sequence of functions where each iteration approaches the value 0, so f(x)= 1 on the interval (0,1) and then a continuous transformation is applied with each interation st at n=infinity f(x)=0. Obviously the function is bounded, but the integral of the function is only pointwise convergent, not uniformly.
