Prove that if f is nondecreasing on ab then f is integrable

Prove that if f is nondecreasing on [a,b] then f is integrable on [a,b]

Solution

Since f(a) ? f(x) ? f(b) for all x ? [a, b], f is bounded.

Given epsilon > 0 there exists k > 0 such that k[f(b)?f(a)] < epsilon .

Let P = {x0, x1, . . . , xn} be a partition of [a, b] such that ?xi < k for all i.

Since f is nondecreasing we have mi = f(xi?1) and Mi = f(xi).

Thus U(f, P)?(f, P) = P(Mi?mi)?xi < k Pf(xi)?f(xi?1) = k[f(b) ? f(a)] < epsilon. Therefore f is integrable.