Chapter 7
Dierentiation
7.1
The Derivative at a Point
Denition 7.1.1. Let f be a function dened on a neighborhood of x0 . f is
dierentiable at x0 , if the following limit exists:
f (x0 ) = lim
h0
f (x0 + h) f (x0 )
.
h
Dene D(f ) = cfw_x : f (x) exists.
T
Chapter 1
Basic Ideas
In the end, all mathematics can be boiled down to logic and set theory. Because
of this, any careful presentation of fundamental mathematical ideas is inevitably
couched in the language of logic and sets. This chapter denes enough of
Chapter 6
Limits of Functions
6.1
Basic Denitions
Denition 6.1.1. Let D R, x0 be a limit point of D and f : D R. The
limit of f (x) at x0 is L, if for each > 0 there is a > 0 such that when x D
with 0 < |x x0 | < , then |f (x) L| < . When this is the case
Chapter 5
The Topology of R
5.1
Open and Closed Sets
Denition 5.1.1. A set G R is open if for every x G there is an > 0
such that (x , x + ) G. A set F R is closed if F c is open.
The idea is that about every point of an open set, there is some room insid
Chapter 3
Sequences
3.1
Basic Properties
Denition 3.1.1. A sequence is a function a : N R.
Instead of using the standard function notation of a(n) for sequences, it is
usually more convenient to write the argument of the function as a subscript,
an .
Exam
Chapter 4
Series
Given a sequence an , in many contexts it is natural to ask about the sum of all
the numbers in the sequence. If only a nite number of the an are nonzero, this
is trivialand not very interesting. If an innite number of the terms arent
zer
NOTES ON REAL ANALYSIS
Math 321
David A. Singer
1. Infinite Series in R
We will consider innite series and their convergence properties. The
results discussed below are largely the work of the nineteent century
mathematician Augustin Louis Cauchy (1789185
Chapter 2
The Real Numbers
This chapter concerns what can be thought of as the rules of the game: the
axioms of the real numbers. These axioms imply all the properties of the real
numbers and, in a sense, any set satisfying them is uniquely determined to
NOTES ON REAL ANALYSIS
Math 321
David A. Singer
1. Continuous Functions
Denition 1.1. Let (X, d) and (Y, d ) be metric spaces. Let a X.
A function f : X Y is continuous at a if for every > 0 there is a
> 0 such that if d(x, a) < , then d (f (x), f (a) <
Math 321
Some Problems for Review
November 25, 2013
2
n
(1) Convergence Verify that the sequence cfw_xn = 3n21 | 1 n con4
1
verges to 3 by using the denition of limit (denition 9.3 on page 70).
(2) Cauchy sequences In any metric space (M, d) a Cauchy seq
B U Department of Mathematics
Fall 2008 Math 321 - Algebra, First Midterm Exam, 14/11/2008, 17:00-19:00
I hope you enjoy the exam!
Full Name Student ID
: :
over 100
In what follows, Zn is the additive group with the modular addition; G, H and K are groups