Page 154 - DMTH401_REAL ANALYSIS

Real Analysis Richa Nandra, Lovely Professional University

Notes Unit 11: Continuity

CONTENTS
Objectives
Introduction

11.1 Continuous Functions
11.2 Algebra of Continuous Functions
11.3 Non-continuous Functions

11.4 Summary
11.5 Keywords
11.6 Review Questions
11.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Define the continuity of a function at a point of its domain
 Determine whether a given function is continuous or not
 Construct new continuous functions from a given class of continuous functions

Introduction

The values of a function f(x) approaching a number A as the variable x approaches a given point
a. When there is break (or jump) in the graph, then this property fails at that point. This idea of
continuity is, therefore, connected with the value of lim f(x) and the value of the function f at the
-
x a
point a. We define in this unit the continuity of a function at a given point a in precise mathematical
language. Therefore extend it to the continuity of a function on a non-empty subset of the
domain of f which could be the whole of the domain of f also. We study the effect of the algebraic
operations of addition, subtraction, multiplication and division on continuous functions.
Here we discuss the properties of continuous functions and the concept of uniform continuity.

11.1 Continuous Functions

We have seen that the limit of a function f as the variable x approaches a given point a in the
domain of a function f does not depend at all on the value of the function at that point a but it
depends only on the values of the function at the points near a. In fact, even if the function f is not
defined at a then lim f(x) may exist.
x a
-
For example lim f(x) exists when
x – 1
2
x - 1
f(x) = though f is not defined at x = 1.
x 1
-
We have also seen that lim f(x) may exist, still it need not be the same as f(a) when it exists.
x a
-
Naturally, we would like to examine the special case when both lim f(x) and f(a) exist and are
x a
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