Page 165 - DMTH401_REAL ANALYSIS

Richa Nandra, Lovely Professional University Unit 12: Properties of Continuous Functions

Unit 12: Properties of Continuous Functions Notes

CONTENTS
Objectives
Introduction
12.1 Continuity on Bounded Closed Intervals
12.2 Pointwise Continuity and Uniform Continuity

12.3 Summary
12.4 Keywords
12.5 Review Questions
12.6 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss the properties of continuous functions on bounded closed intervals

 Explain the important role played by bounded closed intervals in Real Analysis
 Describe the concept of uniform continuity and its relationship with continuity
Introduction

Having studied in the last two units you have studied about limit and continuity of a function at
a point, algebra of limits and continuous functions, the connection between limits and continuity,
etc., we now take up the study of the behaviour of continuous functions and bounded closed
intervals on the real line. You will learn that continuous functions on such intervals are bounded
and attain their bounds; they take all values in between any two values taken at points of such
intervals. You will also be introduced to the concept of uniform continuity and further you will
see that a continuous function on a bounded closed interval is uniformly continuous. This means
that continuous functions are well-behaved on bounded closed intervals. Thus, we will see that
bounded closed intervals form an important subclass of the class of subsets of the real line which
are known as compact subsets of the real line. You will study more about this in higher
mathematics at a later stage. We will henceforth call bounded closed intervals of R as compact
intervals.
The results of this unit play an important and crucial role in Real Analysis and so for further
study in analysis, you must understand clearly the various theorems given in this unit.
It may be noted that an interval of R will not be a compact interval if it is not a bounded or closed
interval.

12.1 Continuity on Bounded Closed Intervals

We now consider functions continuous on bounded closed intervals. They have properties
which fail to be true when the intervals are not bounded or closed. Firstly, we prove the properties
and then with the help of examples we will show the failures of these properties. To prove these
properties, we need an important property of the real line that was discussed in Unit 1.

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