Math 5307 Additional Problems Chapter 7                          Fall 2003

Dr. Cordero

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

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

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

3. Let     and .

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

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

[0, 1]  and find

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

2. 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 .

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

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

2. Evaluate the following integrals:
i.  .
ii.
iii.
iv.
v.
3. 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].

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

1. 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.

1. 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.

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

Hint: Let .

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

4. Find the total variation function v if
i. f(x)=sin x on ,

ii.
5. 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].