Page 177 - DMTH401_REAL ANALYSIS

Richa Nandra, Lovely Professional University Unit 13: Discontinuities and Monotonic Functions

Unit 13: Discontinuities and Monotonic Functions Notes

CONTENTS
Objectives
Introduction

13.1 Discontinuous Functions
13.2 Classification of Discontinuities
13.3 Monotone Function

13.4 Discontinuities of Monotone Functions
13.5 Discontinuities of Second Kind
13.6 Summary
13.7 Keywords
13.8 Review Questions

13.9 Further Readings

Objectives

After studying this unit, you will be able to:
 Define Discontinuous Functions
 Describe Classification of Discontinuities

 Explain Monotone Function
 Describe the Discontinuities of Monotone Functions

 Discuss the Discontinuities of Second Kind
Introduction

In mathematics, a monotonic function (or monotone function) is a function that preserves the
given order. This concept first arose in calculus, and was later generalized to the more abstract
setting of order theory. In calculus, a function f defined on a subset of the real numbers with real
values is called monotonic (also monotonically increasing, increasing or non-decreasing), if for
all x and y such that x d” y one has f(x) d” f(y), so f preserves the order. Likewise, a function is
called monotonically decreasing (also decreasing or non-increasing) if, whenever x d” y, then
f(x) e” f(y), so it reverses the order.

13.1 Discontinuous Functions

If a function fails to be continuous at a point c, then the function is called discontinuous at c, and
c is called a point of discontinuity, or simply a discontinuity.

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