finished in that case. Therefore, we may assume that . In this case, it is possible that one of the is not contained in any . Instead, it may stretch across two of the intervals determined by . (It cannot meet three intervals because is assumed to be smaller than the length of any one interval.) In symbols, it may happen that (We may assume that all the inequalities are strict because otherwise we are in the previous case by our assumption on the length of .) This can happen at most times. To handle this case, we will estimate the difference between the Riemann sum and the Darboux sum by subdividing the partition at . The term in the Riemann sum splits into two terms: Suppose, without loss of generality, that . Then so this term is bounded by the corresponding term in the Darboux sum for . To bound the other term, notice that It follows that, for some (indeed any) , Since this happens at most times, the distance between the Riemann sum and a Darboux sum is at most . Therefore, the distance between the Riemann sum and is at most . Examples Let be the function which takes the value 1 at every point. Any Riemann sum of on will have the value 1, therefore the Riemann integral of on is 1. Let be the indicator function of the rational numbers in ; that is, takes the value 1 on rational numbers and 0 on irrational numbers. This function does not have a Riemann integral. To prove this, we will show how to construct tagged partitions whose Riemann sums get arbitrarily close to both zero and one. To start, let and be a tagged partition (each is between and ). Choose . The have already been chosen, and we can't change the value of at those points. But if we cut the partition into tiny pieces around each , we can minimize the effect of the . Then, by carefully choosing the new tags, we can make the value of the Riemann sum turn out to be within of either zero or one. Our first step is to cut up the partition. There are of the , and we want their total effect to be less than . If we confine each of them to an interval of length less than , then the contribution