This is spivak's calculus 2nd edition #12b. The question in part a defines Pn as a partition of [a,b] into n equal intervals. The question in part b states: Find an integrable function f on [0,1] and an ε > 0 such that U(f,Pn)-∫0 →1 (f) > ε for all n. I'm made no progress on this at all. Part (a) asks to prove that there is no such ε if f is continuous, and I proved that using uniform continuity. I've been trying to create a function that has little jumps at all the rational numbers and none anywhere else to make a counterexample, but that gets me nowhere. Also, I learned that apparently there are different types of integrals, so I think that this book uses the Riemann Stieltjes Integral.