4.6. EXERCISES 59

Hint: Define F (x)≡∫ x

a f (t)g(t) dt and let G(x)≡∫ x

a g(t) dt. Then use the Cauchymean value theorem on these two functions.

8. Consider the function

f (x)≡{

sin( 1

x

)if x ̸= 0

0 if x = 0.

Is f Riemann integrable? Explain why or why not.

9. Prove the second part of Theorem 4.3.2 about decreasing functions.

10. Suppose f is a bounded function defined on [a,b] and | f (x)| < M for all x ∈ [a,b] .Now let Q be a partition having n points,

{x∗0, · · · ,x∗n

}and let P be any other partition.

Show that

|U ( f ,P)−L( f ,P)| ≤ 2Mn ||P||+ |U ( f ,Q)−L( f ,Q)| .

Hint: Write the sum for U ( f ,P)− L( f ,P) and split this sum into two sums, thesum of terms for which [xi−1,xi] contains at least one point of Q, and terms forwhich [xi−1,xi] does not contain any points of Q. In the latter case, [xi−1,xi] must becontained in some interval,

[x∗k−1,x

∗k

]. Therefore, the sum of these terms should be

no larger than |U ( f ,Q)−L( f ,Q)| .

11. ↑ If ε > 0 is given and f is a Darboux integrable function defined on [a,b], showthere exists δ > 0 such that whenever ||P||< δ , then

|U ( f ,P)−L( f ,P)|< ε.

12. ↑ Prove Theorem 4.5.6.