1549901369-Elements_of_Real_Analysis__Denlinger_

7.4 Basic Existence and Additivity Theorems 395

Corollary 7.4.19 If f is a regulated function on [a, b], then f is integrable on

[a,b].

Proof. Suppose f is regulated on [a, b] with a< b. Let E: > 0. By Theorem

7.4.18, 3 step function a : [a, b] --+JR. 3 \:Ix E [a, b],

E:

lf(x) - a(x)I < 4 (b _a)

E: E:

a(x) - 4(b - a) < f(x) < a(x) + 4(b - a).

Define step functions T 1 , T 2 : [a, b] --+ JR. by

E: E:

T^1 (x)=a(x)- 4 (b-a) and T2(x)=a(x)+ 4 (b-a)

Then \:Ix E [a, b], T 1 (x) < f(x) < T2(x) and

b b E: c(b-a)

{^72 -^71 = { 2(b - a) = 2(b - a) < E:.

Therefore, by Theorem 7.4.14, f is integrable on [a, b]. •

Corollary 7.4. 20 Thomae' s function is integrable on every compact interval.

Not all integrable functions are regulated. For an example of a function
that is integrable on [a, b] but not regulated there, see Exercise 17.

EXERCISE SET 7.4

1. Prove Lemma 7.4.l (b).


2. Prove Corollary 7.4.3. [Use mathematical induction.]


3. Prove Corollary 7.4.4.


4. How do Theorems 7.4.2 and 7.4.5 differ?


5. Prove Corollary 7.4.6. [Use mathematical induction.]


6. Prove that if f is integrable on [a, b], then 3M > 0 3 for all subintervals
   [c, d] ~ [a, b], Jt f I :S M(d - c). [See Exercise 7.2.3.]


7. Prove the remaining equalities in Theorem 7.4.7 (b).


8. Prove Theorem 7.4.13.


9. Determine whether each of the following functions is integrable over
   [-1, l]:

(a) f ( x) = { sin ( ~) ~f x =I 0 }

0 ifx=O

(b) g( x) = { ~ sin x if x =I 0 }

0 if x = 0