318 CHAPTER 13. THE GENERALIZED RIEMANN INTEGRAL

13.5 Exercises1. Prove that if fn ∈R∗ [a,b] and { fn} converges uniformly to f , and | fn− fm| ∈R∗ [a,b]

for each m,n, then f ∈ R∗ [a,b] and limn→∞

∫I fn =

∫I f .

2. Suppose the integrator function is F (x) = x. Show that for I any interval, XI isRiemann Stieltjes integrable and if I ⊆ [a,b] , then

∫ ba XIdx is the length of I.

3. In Example 13.4.5 there is the function given

g(x)≡

{2xsin

(1x2

)− 2

x cos(

1x2

)if x ̸= 0

0 if x = 0

It equals the derivative of a function as explained in this example. Thus g is general-ized Riemann integrable on [0,1]. Show that h(x) = max(0,g(x)) and h(x) = |g(x)|are not generalized Riemann integrable.

4. Let f ∈R∗ [a,b] and consider the function x→∫ x

a f (t)dt. Is this function continuous?Explain. Hint: Let ε > 0 be given and let a gauge δ be such that if P is δ fine then∣∣∣∣S (P, f )−

∫ b

af dx∣∣∣∣< ε/2

Now pick h < δ (x) for some x ∈ (a,b) such that x+h < b. Then consider the singletagged interval, ([x,x+h] ,x) where x is the tag. By Corollary 13.1.15∣∣∣∣ f (x)h−

∫ x+h

xf (t)dt

∣∣∣∣< ε/2.

Now you finish the argument and show f is continuous from the right. A similarargument will work for continuity from the left.

5. Generalize Problem 4 to the case where the integrator function is continuous. Whatif the integrator function is not continuous at x? Can you say that continuity holds atevery point of continuity of F?

6. If F is a real valued increasing function, show that it has countably many points ofdiscontinuity.

7. If C ≡ {ri}∞

i=1 is a countable set in [a,b] , show that XC is in R∗ [a,b] . Hint: LetCn = {r1, · · · ,rn} and explain why XCn is generalized Riemann integrable. Then usethe monotone convergence theorem.

8. Prove the first mean value theorem for integrals for the generalized Riemann integralin the case that x→

∫ xa f (t)dF is continuous.

9. Suppose f , | f | ∈ R∗ [a,b] and f is continuous at x ∈ [a,b] . Show G(y)≡∫ y

a f (t)dt isdifferentiable at x and G′ (x) = f (x).

10. Suppose f has n+1 derivatives on an open interval containing c. Show using induc-tion and integration by parts that

f (x) = f (c)+n

∑k=1

f k (c)k!

(x− c)k +1n!

∫ x

cf (n+1) (t)(x− t)n dt.