real analysis

can you look at the attachments?
it is chapter about integral.

10. Suppose F is bounded on [a,b].

(a) Prove that for any partition P of [a,b] and any Riemann sum (f), we have

(b) Suppose f is continuous on [a,b] and >0. Because f is uniformly continuous on [a,b], there is a >0 such that

when <. Prove that for any partition P with and any associated Riemann sum . (Hint: Use the proof of “If F is continuous on [a,b], then it is integrable on [a,b])

(c) Assume that F is continuous on [a,b]. Show that exists and is equal to