Math 327B
Homework #1
due Wednesday, October 7
1
Homework assignments in this class will be usually due on Wednesday. The importance of doing
all the homework cannot be overemphasized. Mathematics cannot be learned passively. Even professional mathematici
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Problem 1. Let f be a function defined on a closed domain. Show that f is continuous if
and only if the inverse image of every closed set is a closed set.
Solution. Now assume f is continuous. Let D be a closed set, then we need to prove that
f 1 (D) is a
Advanced Calculus I
Chapter 3 Homework Solutions
November 20, 2009
First Assignment
27. Prove that every set of the form cfw_ x | a < x < b is open and every set of the form
cfw_ x | a x b is closed.
Pick x (a, b). Set = 12 mincfw_xa, bx, and note that
Hw 3
Read chapter 2 of the textbook.
Main skills:
You need to know the definition of limit of a sequence being
You need to know the definition of monotone and bounded sequence and
relevant theorems.
You need to know the definition of subsequence of a
Hw 5
Read chapter 3 of the textbook.
Main skills:
You need to know the definition of limit of a function.
You need to know the definition of continuous function
Do the following problems:
1. Let f : D R and g : D R be two functions, x0 be a limit point
35
6
Continuous functions
6.1
Definition and examples
All this work about limits will now pay off since we shall be able to give mathematically rigorous
definition of a continuous function.
Definition 6.1.1. Let f be a real valued function of a real varia
Hw 6
Read chapter 3 of the textbook.
Main skills:
You need to know the extreme value and the intermediate value theorem.
You need to know the definition of uniform continuity
Do the following problems:
1. Find the image of the following functions :
(a)
Hw 4 Part 1
Read chapter 9.1 of the textbook.
Main skills:
You need to know the definition of series.
You need to know about geometric and harmonic series
You need to know the comparison test
You need to know the limit test
Do the following problems:
Hw 1
Read chapter 1 of the textbook.
Main skills:
You need to know the axioms defining R.
You need to be understand the definition of sup S and inf S and be able
to calculate sup S and inf S.
. Do the following problems.
1. For each of the following sub
Hw 4 Part 2
Read chapter 9.1 of the textbook.
Main skills:
You need to know what absolute convergence is
You need to know the ratio test
You need to know the alternating series test
Do the following problems:
1. Decide if the following series are conve
Hw 2
Read chapter 2 of the textbook.
Main skills:
You need to review induction
You need to know some useful inequalities and identities.
You need to know the definition of limit of a sequence, and be able to use
it in proofs.
You need to know the limi
Math 327B
Solutions for Homework #1
October 7, 2015
1
1. Find the sup, inf, max, and min of each of the following sets, if it exists. Prove your answers!
(a) A = [2, 8) = cfw_x R : 2 x < 8
We claim that inf A = 2. To prove this, we first note that by defi
Math 327: Series in R
Mary Radclie
In this set of notes, we explore series in R, a special kind of sequence. We
develop theorems related to convergence and limits.
1
Introduction and Denitions
In the previous set of notes, we developed theory related to s