1549901369-Elements_of_Real_Analysis__Denlinger_

7.9 *Lebesgue's Criterion for Riemann Integrability 443

(c)

1

(n+l)7r I sin xi n l(k+l)?r I sin x i

Show that 'Vn EN, · --dx = L --dx.

7r x k=l k?r x

(d) Showthat'VkEN,'VxE [br+~, (k+l)K-~], lsinxl:::::^1.

x 2(k + l)7r

1

(n+l)7r I sin x i

(e) Use (c) and (d) to prove that lim - - dx = +oo. [Recall

n---too 7r X

the harmonic series, Example 2.5.16.J

6. One might expect that if J 000 f converges, then 'Vx > 0, lim J(x) = 0,
   x->oo
   but this is not necessarily true, as shown by J 0
   00
   cos x^2 dx.

(a) Use integration by parts to obtain J cos x^2 dx = sin x

2

+ j sin x

2

2

dx.

2x 2x

(b) Use the result of (a) to prove that J 0

00

cos x^2 dx converges. Show that

X->00 lim cos x^2 does not exist.

7. Prove that in Definition 7.8. 11 the choice of c affects neither the conver-
   gence of J~:; f nor its value.


8. (a) Prove^22 Theorem 7.8.13.
   (b) State and prove a modification of Theorem 7.8.13 in which "O <
   f(x)::; g(x)" is replaced by "O::::: f(x)::::: g(x)." Illustrate graphically.


9. (a) Prove Theorem 7.8.14.
   (b) State and prove a modification of Theorem 7.8.14 in which "O <
   f(x) ::; g(x)" is replaced by "O::::: f(x) ::::: g(x)." Illustrate graphically.

10. Prove Theorem 7.8.16.

7.9 * Lebesgue's Criterion for Riemann

Integrability

This section can be skipped or assigned as independent read-

ing. The concepts are abstract and the proofs are challenging.

In this section we present what is perhaps the most celebrated criterion for the
integrability of a bounded function on a compact interval. It is a straightfor-
ward criterion, involving only the set of points of discontinuity of the function.
This truly remarkable criterion is easy to state but far from easy to prove. We
begin by stating the criterion as a theorem. Then we develop its proof in stages.

22. Exercise 4.4-B.15 will be helpful in this and the following exercise.