Calculus: Analytic Geometry and Calculus, with Vectors

5.7 Darboux sums and Riemann integrals 353

the interval a<= t =< x. Prove that T(x), P(x), N(x) are all nonnegative and
monotone increasing over a< x < b. Prove that

(4)

(5)

(6)

(7)

(8)

T(x) = 2P(x) - [f(x) - f(a)]

T(x) = 2N(x) + [f(x) -f(a)]

T(x) = P(x) + N(x)

f(x) = f(a) + P(x) - N(x)

f(x) _ [x + f(a) + P(x)] - [x + N(x)]

when a < x < b. Use our information to prove that if f (x) has bounded variation

over a < x < b [which means that T(b) is finite], then f(x) is the difference of two

increasing functions. Prove that if f(x) is the difference of two increasing func-

tions over a < x < b, then f(x) has bounded variation over a S x 5 b. Hint:

To obtain (4), note that (2) contains a telescopic sum and put (2) in the form

(9)^1

n

l.u.b. f (x) - f (a)

I 1If(tk) -f(tk-1)I + 2 }= P(x).

Remark: Our results and Theorem 5.76 imply that the Riemann integral fab f (x) dx

exists if f has bounded variation over a < x 5 b. Moreover, we now have enough

information to appreciate the most important theorem in the theory of Riemann-

Stieltjes integrals; see Problem 11 of Problems 4.89. The theorem says that

(10) fabf(x) dg(x)

exists if f is continuous and g has bounded variation over a < x < b. Methods

of this section provide proof for the case in which f is continuous and g is increas-

ing, and the general result is then obtained by expressing g as the difference of

increasing functions. A much more difficult theorem says that if g is such that

(10) exists whenever f is continuous over a - x < b, then g must have bounded

variation over a 5 x < b.