Math 5307 Additional Problems Chapter 7 Fall 2003

Dr. Cordero

In Problems 1-10 f is bounded and is monotone increasing on [a, b].

- Let f(x)=c on [a, b].
Show that on [a, b] and
.

- Let
(x)=c on [a, b]. Show that
on [a, b] and
.

- Let
and .

Show that f is not in R() on [0, 1].

- Let and let be defined as in Exercise 3. Show that g is in R() on

[0, 1] and find

- If f is continuous and
nonnegative on [a, b] with (a)< (b), show that there is a c in [a, b] such that
.

- If f is continuous on
[a, b] and g is in R() on [a, b] with g nonnegative, show that there is a c in
[a, b] such that .

- Let f be monotone and
continuous on [a, b]. Show that there is a c in [a, b]
such that .

- Suppose f and
are monotone increasing on [a, b] and P is a partition
of [a, b]. Prove that .

- Evaluate the following
integrals:

i. .

ii.

iii.

iv.

v. - Suppose
on [a, b]. For
define .

i. Prove that for some positive constant M and all x, y in [a, b].

ii. Prove that every point of continuity of is a point of continuity of F.

iii. Prove that if either (1) f is continuous and is of bounded variation on [a, b], or (2) f and are both of bounded variation and is continuous on [a, b], then F is of bounded variation on [a, b].

- Evaluate where f is bounded on [-1, 1] and continuous at 0, and is given by:

- Suppose f is Riemann
integrable on [a, b] and g is a bounded real-valued function on [a, b]. If
has measure zero, is g Riemann integrable on [a, b]?
Prove or give a counterexample.

__Additional problems Chapter 6__

- i. Show that
if and only if f is a constant function on [a, b].

ii. Show that |f| is of bounded variation on [a, b] if f is of bounded variation on [a, b].

iii. Give an example showing that |f| being of bounded variation need not imply that f is of bounded variation.

- Show that
is continuous but not of bounded variation on [0,
1].

Hint: Let .

- Show that
is of bounded variation on [0,1].

- Find the total
variation function v if

i. f(x)=sin x on ,

ii. - Show that if a function f satisfies a Lipschitz condition of order 1 on [a, b], then f is of bounded variation on [a, b]. Give an example of a function of bounded variation that satisfies no Lipschitz condition on [a, b].