1549901369-Elements_of_Real_Analysis__Denlinger_

396 Chapter 7 • The Riemann Integral

{

lx

2

• ll .f =J 1 }

10. Draw the graph of the function f(x) = x - 1

1

x on the in-

0 if x = 1

terval [-2, 2]. Use theorems from this section to prove that f is integrable

over [-2, 2], and use your knowledge of integral as "area" to find f~ 2 f.

11. Let l x J = the greatest integer ::; x, the so-called "greatest integer func-
    tion." Use geometry to determine the existence, and the value, of each
    of the following:

(a) f~ lxJ dx (b) f~ x - lxJ dx (c) f~ 2 x lxJ dx

12. Suppose f is integrable and nonnegative (or nonpositive) on [a, b]. Prove

that l:f[c, d] i;;;; [a, b], J: f ::; f: f (or J: f ~ f: f in the non positive case).

1 3. Suppose f is bounded on [a,b] and P = {xo,x1,x2, · · · ,xn} is a par-

tition of [a,b]. Let mi and Mi be as in Definition 7.2.1 and let XA de-

note the characteristic function of a set A. Define u , T : [O, 1] ---> JR by

n n

u(x) = L ffiiX[x,_i,x;)(x) + f(b)x{b}(x) and T(x) = L MiX[x,_ 1 ,x,)(x) +

i=l i=l

f(b)X{b}(x). Show that u and Tare step functions and \ix E [a, b], u(x)::;

f(x)::; T(x). Also show that f: u = Q.(j, P) and f: T = S(f, P).

14. Prove Theorem 7.4.14. [Hint: Show t hat this theorem is a rewording of

Riemann's condit ion (7.2.14). Use Exercise 13 as needed.]

15. Find a function f that is integrable on every closed subinterval of (0, 1)

but that is not integrable on [O, l]. Does this contradict Theorem 7.4.7?

16. Finish P art 2 of the proof of Theorem 7.4.18.

17. Show that the funct10n. f(x) = { sin x l if^0 < x -< 1; }

0 if x = 0

is integrable on

[O, l] but is not regulated on [O, l].

1 8. Find f 01 T of Thomae's function, defined in Example 5.1.12, as follows:

l:fc > 0, define 0" 0 : [O, 1] ---> JR by u 0 (x) = max{T(x), c }. Prove that u 0

is a step function on [O, l], and observe that 0 ::; f 01 T ::; f 01 u 0 ::; c. [See

Exercise 7.2. 1 3.]

Note: this is an example of a function that is integrable on [O, 1] yet

discontinuous on a dense subset of [O, l].

19. Use Theorem 5.7.3 to prove that \fa< b, if A is any countable dense subset

of [a, b], t here is a function t hat is integrable on [a, b] yet is discontinuous

on A.