Chapter 7 – Additional Problems

Math 5307 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 
5. If f is continuous and nonnegative on [a, b] with  (a)<  (b), show that there is a c in [a, b] such hat .
6. 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 .
7. Let f be monotone and  continuous on [a, b]. Show that there is a c in [a, b] such that .
8. Suppose f and  are monotone increasing on [a, b] and P is a partition of [a, b]. Prove that .
9. Evaluate the following integrals:
   1. 
   2. 
   3. 
   4. 
   5. 
10. Suppose  on [a, b]. For  define .
    1.  Prove that  for some positive constant M and all x, y in [a, b].
    2. Prove that every point of continuity of  is a point of continuity of F.
    3. 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].
11. Evaluate  where f is bounded on [-1, 1] and continuous at 0, and  is given by:
    
12. 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.

Chapter 6 – Additional problems

1. Show that  if and only if f is a constant function on [a, b].
2. Show that |f| is of bounded variation on [a, b] if f is of bounded variation on [a, b].
3. Give an example showing that |f| being of bounded variation need not imply that f is of bounded variation.