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 the cubes of the
rst n natural numbers.
Problem 1. Prov
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) Prove that if g (x) = f (cx), then g (x) = c f (cx).
(
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 + 3 = 0 has a solution between n and
n + 1.
Problem 3.
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 that uniformly is necessary.
Problem 2.
(i) Prove that if
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.
Problem 2. Prove or give a counterexample:
(i) The union of
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 are true: (a) The union
of two uncountable sets is unco
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.
(ii) Find a function f : R R which is continuous only at
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 numbers.
Problem 2. A sequence (an ) is said to be Cauc
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 the cubes of the
rst n natural numbers.
Problem 1. Prov
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) = f (cx), then g (x) = c f (cx).
(iii) Suppose that f
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 that the polynomial equation x3 x + 3 = 0 has a solution b
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 uniformly is necessary.
Solution. If f is uniformly cont
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 compact.
Solution. (i) Let cfw_Ki iI be a family of compa
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 numbers.
Solution. (i) The interval [0, 1]. (ii) The set