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Sachin Kaushal, Lovely Professional University Unit 1: Differentiation and Integration: Differentiation of Monotone Functions

Unit 1: Differentiation and Integration: Differentiation Notes
of Monotone Functions

CONTENTS
Objectives

Introduction
1.1 Differentiation and Integration
1.1.1 Lipschitz Condition
1.1.2 Lebesgue Point of a Function

1.1.3 Covering in the Sense of Vitali
1.1.4 Four Dini's Derivatives
1.1.5 Lebesgue Differentiation Theorem
1.2 Summary

1.3 Keywords
1.4 Review Questions
1.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand differentiation and integration
 Describe Lipschitz condition and Lebesgue point of a function
 State Vitali’s Lemma and understand its proof.
 Explain four Dini’s derivatives and its properties

 Describe Lebesgue differentiation theorem.

Introduction

Differentiation and integration are closely connected. The fundamental theorem of the integral
calculus is that differentiation and integration are inverse processes. The general principle may
be interpreted in two different ways:
1. If f is a Riemann integrable function over [a, b], then its indefinite integral i.e.
x
F : [a, b] R defined by F (x) = f (t) dt is continuous on [a, b]. Furthermore if f is
a
continuous at a point x [a, b], then F is differentiable thereat and F (xo) = f (x ).
o o
2. If f is Riemann integrable over [a, b] and if there is a differentiable function F on [a, b] such
that F f(x) for x [a, b], then
x
f (t) dt = F (x) – F (a) [a x b].
a

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