It happens to be integrable. First note is bounded since . (Also assume that because otherwise [tex]f[/math[ would be a constant function, and there is nothing to prove).

Let be the upper Darboux sum with respect to partition of and be the lower Darboux sum with respect to partition . To show this function is integrable we need to show for any we can find a partition such that . Note, since is monotone it means . Let be . Then we see that . Thus, if then . This means given any we can create equal subdivision of the interval until the interval length and then we are have . Thus is integrable.