Page 217 - DMTH401_REAL ANALYSIS

Sachin Kaushal, Lovely Professional University Unit 17: Uniform Convergence and Differentiability

Unit 17: Uniform Convergence and Differentiability Notes

CONTENTS
Objectives
Introduction
17.1 Uniform Convergence and Differentiability
17.2 Series of Functions
17.3 Central Principle of Uniform Convergence

17.4 Power Series and Uniform Convergence
17.5 Continuous but Nowhere Differentiable Function
17.6 Summary
17.7 Keywords
17.8 Review Questions
17.9 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss the uniform convergence and differentiability

 Explain the series of the functions
 Describe the central principle of uniform convergence
Introduction

In last unit you have studied about uniform convergence and continuity. Simple or pointwise
convergence is not enough to preserve differentiability, and neither is uniform convergence by
itself. However, if we combine pointwise with uniform convergence we can indeed preserve
differentiability and also switch the limit process with the process of differentiation. It remains
to clarify the connection between uniform convergence and differentiability.

17.1 Uniform Convergence and Differentiability

Is it true that if all f are differentiable and f ® f uniformly then f is differentiable and
n n
¢
f ® f ? ¢ The answer is no. Look at the following examples.
n
1
2
Example: Let f n :  ®  , f n ( ) = sin(n x .
)
x
n
æ 1 2 ö
)
We see that all f are differentiable and that f ® f uniformly ç è f - f sup = n sin(n x = 1/n ® 0 .
÷
n
n
n
ø
æ 1 2 ö
)
2
=
x
)
0
ç f - f = sin(n x = 1/n ® 0 . But f ¢ ( ) n cos(n x ® neither uniformly nor pointwise, although the zero function
÷
n
è sup n ø n
is differentiable.
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