Kenneth Kuttler

56 CHAPTER 4. THE RIEMANN STIELTJES INTEGRAL

and this proves the theorem.Note this gives existence for the initial value problem,

F ′ (x) = f (x) , F (a) = 0

whenever f is Riemann integrable and continuous.3

The next theorem is also called the fundamental theorem of calculus.

Theorem 4.5.2 Let f ∈ R([a,b]) and suppose there exists an antiderivative for f ,G, suchthat

G′ (x) = f (x)

for every point of (a,b) and G is continuous on [a,b] . Then∫ b

af (x) dx = G(b)−G(a) . (4.5.16)

Proof: Let P = {x0, · · · ,xn} be a partition satisfying

U ( f ,P)−L( f ,P)< ε.

Then

G(b)−G(a) = G(xn)−G(x0)

∑i=1

G(xi)−G(xi−1) .

By the mean value theorem,

G(b)−G(a) =n

∑i=1

G′ (zi)(xi− xi−1)

∑i=1

f (zi)∆xi

where zi is some point in [xi−1,xi] . It follows, since the above sum lies between the upperand lower sums, that

G(b)−G(a) ∈ [L( f ,P) ,U ( f ,P)] ,

and also ∫ b

af (x) dx ∈ [L( f ,P) ,U ( f ,P)] .

Therefore, ∣∣∣∣G(b)−G(a)−∫ b

af (x) dx

∣∣∣∣<U ( f ,P)−L( f ,P)< ε.

Since ε > 0 is arbitrary, 4.5.16 holds. This proves the theorem.3Of course it was proved that if f is continuous on a closed interval, [a,b] , then f ∈ R([a,b]) but this is a hard

theorem using the difficult result about uniform continuity.

56 CHAPTER 4. THE RIEMANN STIELTJES INTEGRALand this proves the theorem.Note this gives existence for the initial value problem,F'(x) = f(x), F(a) =0whenever f is Riemann integrable and continuous.*The next theorem is also called the fundamental theorem of calculus.Theorem 4.5.2 Let f € R(|a,b]) and suppose there exists an antiderivative for f,G, suchthatG' (x) = f (x)for every point of (a,b) and G is continuous on [a,b]. Thenb| f (x) dx =G(b) —G(a). (4.5.16)aProof: Let P = {xo,--- ,x,} be a partition satisfyingU(f,P)-L(f,P) <E.ThenBy the mean value theorem,Ms=Q—NXQa&st|a,G(b)-G(a) =)ll=TM:=Ewhere z; is some point in [x;—1,.;] . It follows, since the above sum lies between the upperand lower sums, thatG(b) —G(a) € [L(f,P),U(f,P)],and also b[ fave LP) UP).Therefore,eo) -Ga- [1 dx| <U(f,P)—L(f,P) <€.Since € > 0 is arbitrary, 4.5.16 holds. This proves the theorem.3Of course it was proved that if f is continuous on a closed interval, [a,b], then f € R([a,b]) but this is a hardtheorem using the difficult result about uniform continuity.