Page 210 - DMTH401_REAL ANALYSIS

Real Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 16: Uniform Convergence and Continuity

CONTENTS
Objectives
Introduction
16.1 Uniform Convergence Preserves Continuity
16.2 Uniform Convergence and Supremum Norm
16.3 Uniform Convergence and Integrability

16.4 Convergence almost Everywhere
16.5 Lebesgue’s Bounded Convergence Theorem
16.6 Summary
16.7 Keywords
16.8 Review Questions
16.9 Further Readings

Objectives

After studying this unit, you will be able to:
 Define Uniform Convergence preserves Continuity

 Explain the Supremum Norm
 Discuss Sup-norm and Uniform Convergence
Introduction

In earlier unit as you all studied about uniform convergence. Uniform convergence clearly
implies pointwise convergence, but the converse is false. Therefore uniform convergence is a
more “difficult” concept. The good news is that uniform convergence preserves at least some
properties of a sequence. This unit will explain Uniform Convergence preserves Continuity.

16.1 Uniform Convergence Preserves Continuity

If a sequence of functions f (x) defined on D converges uniformly to a function f(x), and if each
n
f (x) is continuous on D, then the limit function f(x) is also continuous on D.
n
All ingredients will be needed, that f converges uniformly and that each f is continuous. We
n n
want to prove that f is continuous on D. Thus, we need to pick an x and show that
0
|f(x ) – f(x)| <  if |x – x| < 
0 0
Let’s start with an arbitrary  > 0. Because of uniform convergence we can find an N such that
|f (x) – f(x)| < /3 if n N
n
for all x D. Because all f are continuous, we can find in particular a  > 0 such that
n
|f (x ) – f (x)| < /3 if |x - x| < 
N 0 N 0

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