> 0 there exists a partition Pε of [a, b] such that Properties For any given partition, the upper Darboux sum is always greater than or equal to the lower Darboux sum. Furthermore, the lower Darboux sum is bounded below by the rectangle of width (b−a) and height inf(f) taken over [a, b]. Likewise, the upper sum is bounded above by the rectangle of width (b−a) and height sup(f). The lower and upper Darboux integrals satisfy Given any c in (a, b) The lower and upper Darboux integrals are not necessarily linear. Suppose that g:[a, b] → R is also a bounded function, then the upper and lower integrals satisfy the following inequalities. For a constant c ≥ 0 we have For a constant c ≤ 0 we have Consider the function: then F is Lipschitz continuous. An identical result holds if F is defined using an upper Darboux integral. Examples A Darboux-integrable function Suppose we want to show that the function is Darboux-integrable on the interval and determine its value. To do this we partition into equally sized subintervals each of length . We denote a partition of equally sized subintervals as . Now since is strictly increasing on , the infimum on any particular subinterval is given by its starting point. Likewise the supremum on any particular subinterval is given by its end point. The starting point of the -th subinterval in is and the end point is . Thus the lower Darboux sum on a partition is given by similarly, the upper Darboux sum is given by Since Thus for given any , we have that any partition with satisfies which shows that is Darboux integrable. To find the value of the integral note that A nonintegrable function Suppose we have the Dirichlet function defined as Since the rational and irrational numbers are both dense subsets of , it follows that takes on the value of 0 and 1 on every subinterval of any partition. Thus for any partition we have from which we can see that the lower and upper Darboux integrals are unequal. Refinement of a partition and relation to Riemann integration A refinement of the partition is a partition such that for all i = 0, …, n there is an integer r(i) such that In other words, to make a refinement, cut the subintervals into smaller pieces and do not remove any existing