Math 512A. Homework 10. Due 12/5/07
b
b4
, by considering upper and lower sums for partitions of [0, b]
4
0
into n equal subintervals, using the formula 13 + 23 + + n3 = (1 + 2 + + n)2 for the sum of
Math 512A. Homework 9. Due 11/14/07
(Revised 11/10)
Problem 1. (i) Suppose that g (x) = f (x + c) for all x. Prove, starting from the denition of the derivative,
that g (x) = f (x + c) for all x.
(ii)
Math 512A. Homework 8. Due 10/7/07
Problem 1. Find the maximum value of f (x) = x3 9x in the interval [3, 3]. Note: No derivatives!
Problem 2. Find an integer n such that the polynomial equation x3 x
Math 512A. Homework 7. Due 10/31/07
Problem 1. Let f : E R R be uniformly continuous. Prove that if (xn ) is a Cauchy sequence in E , then
(f (xn ) is also a Cauchy sequence. Show by counterexample th
Math 512A. Homework 6. Due 10/24/07
Problem 1. Prove the following:
(i) The intersection of an arbitrary family of compact sets is compact.
(ii) The union of nitely many compact sets is compact.
Probl
Math 512A. Homework 5. Due 10/10/07
(Work out any 5 problems)
(Revised 10/4)
Problem 1.
(i) Dene countable set.
(ii) Determine (either prove or give a counterexample) whether the following statements
Math 512A. Homework 4. Due 10/3/07
(Revised 9/30)
Problem 1. (You must show that your example satises the required property.)
(i) Find a function f : R R which is continuous except at the integers.
(i
Math 512A. Homework 3. Due 9/19/07
Problem 1. Find all the accumulation points of the following sets:
(i) The interval [0, 1).
(ii) The set of all the irrational numbers.
(iii) The set of the natural
Math 512A. Homework 10. Solutions
b
b4
, by considering upper and lower sums for partitions of [0, b]
4
0
3
3
into n equal subintervals, using the formula 1 + 2 + + n3 = (1 + 2 + + n)2 for the sum of
Math 512A. Homework 9. Solutions
Problem 1. (i) Suppose that g (x) = f (x + c) for all x. Prove, starting from the denition of the derivative,
that g (x) = f (x + c) for all x.
(ii) Prove that if g (x
Math 512A. Homework 8 Solutions
Problem 1. Find the maximum value of f (x) = x3 9x in the interval [3, 3]. Note: No derivatives!
Solution. This was done in class.
Problem 2. Find an integer n such tha
Math 512A. Homework 7 Solutions
Problem 1. Let f : E R R be uniformly continuous. Prove that if (xn ) is a Cauchy sequence in E , then
(f (xn ) is also a Cauchy sequence. Show by counterexample that u
Math 512A. Homework 6 Solutions
(Revised 10/27)
Problem 1. Prove the following:
(i) The intersection of an arbitrary family of compact sets is
compact.
(ii) The union of nitely many compact sets is co
Math 512A. Homework 3. Solutions
Problem 1. Find all the accumulation points of the following sets:
(i) The interval [0, 1).
(ii) The set of all the irrational numbers.
(iii) The set of the natural nu