WY Wong /SL Tan UECM2033 Elementary Real Analysis
Chapter 4
The Derivative
4.1 Introduction
The derivative is a measure of how a function changes as the variable in the
domain of the function changes. Loosely speaking, a derivative measures the
changes of
WY Wong/SL Tan
UECM2033 Elementary Real Analysis
UECM 2033 Elementary Real Analysis
Reference book:
1. Walter Rudin, Principle of mathematical analysis, McGraw-Hill
2. Bartle And Sherbert, Introduction to real analysis, John Wiley
Chapter 1
Real Numbers
1
Chapter 5
Integration
5.1 Introduction
There is often confusion among students of calculus between integral and
antiderivative. The integral is the limit of a Riemann sum which gives
(informally) the area under the curve. The antiderivative, say F of a gi
WY Wong/Tan Sin Leng UECM2033
Elementary Real Analysis
Chapter 3
Continuous Functions
3.1 Introduction
We will study the class of continuous functions in this chapter. Works of
Bolzano in 1817 and Cauchy in 1821 laid down some important properties of
cont
WY Wong /SL Tan UECM2033 Elementary Real Analysis
Chapter 7
Topology of R
Introduction
The first six chapters have been devoted primarily to the theory for single
variable calculus. The functions have been, for the most part, of a type
familiar to us from
WY Wong /SL Tan UECM2033 Elementary Real Analysis
Chapter 6
Sequences of Functions
6.1 Introduction
In Chapter 2, we have talked about sequences of numbers, how they converge
or diverge. In this chapter, we will deal with convergence of sequence of
functi
Faculty:
Course:
Year:
Session:
Tutorial 1
FES
AM/FM
Two/Three
Jan. 2016
Unit Code:
UECM2033
Unit Title:
Elementary Real Analysis
Lecturer/Tutor: Tan Sin Leng/Tan Sin Leng
0. (a) If we have two real numbers a and b, how do we know that they are not equal?
Faculty:
Course:
Year:
Session:
Tutorial 2
FES
AM/FM
Two/Three
Jan. 2016
Unit Code:
UECM2033
Unit Title:
Elementary Real Analysis
Lecturer/Tutor: Tan Sin Leng/Tan Sin Leng
1
1. For n N , show that cos
n
sin
1
n
1
n 1.
2. For the two open intervals I = (-
Faculty:
Course:
Year:
Session:
LKC FES
AM/FM
Two/Three
Jan. 2016
Unit Code:
UECM2033
Unit Title:
Elementary Real Analysis
Lecturer/Tutor: Tan Sin Leng/Tan Sin Leng
Tutorial 3
1. (a) What can be said about the sequence an if it converges and each an is an
WY Wong/Tan Sin Leng UECM2033
Elementary Real Analysis
Chapter 2
Sequences and Series
2.1 Sequences
Definition 2.1.1 (Sequence)
A sequence is a function x : N R .
Instead of writing the image of x as x(n) , we denote the image of n under
x as xn .
We us