(a) If f is increasing on [a, bJ and P = {xo, ... , xn } is any partition of [a, bJ, prove that n

'L(Mj

(f) - mj(f))(xj - Xj-1) :::; (f

(b) - f(a)) IIPII·

j=l

(b) Prove that if f is monotone on [a,b], then f is integrable on [a,bJ.

[Note: By Theorem 4.29, f has at most countably many (i.e., relatively few)

discontinuities on [a,bJ. This has nothing to do with the proof of part (b), but points out a general principle that will be discussed in Section 9.6.J