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Sachin Kaushal, Lovely Professional University Unit 2: Functions of Bounded Variation

Unit 2: Functions of Bounded Variation Notes

CONTENTS
Objectives

Introduction
2.1 Functions of Bounded Variation
2.1.1 Absolute Continuous Function
2.1.2 Monotonic Function
2.1.3 Functions of Bounded Variation – Definition

2.1.4 Theorems and Solved Examples
2.2 Summary
2.3 Keywords
2.4 Review Questions
2.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Define absolute continuous function.

 Define monotonic function.
 Understand functions of bounded variation.
 Solve problems on functions of bounded variation.

Introduction

Functions of bounded variation is a special class of functions with finite variation over an
interval. In Mathematical analysis, a function of bounded variation, also known as a BV function,
is a real-valued function whose total variation is bounded: the graph of a function having this
property is well behaved in a precise sense. Functions of bounded variation are precisely those
with respect to which one may find Riemann – Stieltjes integrals of all continuous functions.
In this unit, we will study about absolute continuous function, Monotonic function and functions
of bounded variation.

2.1 Functions of Bounded Variation

2.1.1 Absolute Continuous Function

A real-valued function f defined on [a,b] is said to be absolutely continuous on [a,b], if for an
arbitrary 0 , however small, a, 0, such that
n n
f b – f a ,wherever b a ,
r r r r
r 1 r 1

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