Jump Discontinuity

3 Key excerpts on "Jump Discontinuity"

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  eBook - ePub

  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 8 The Concept of Continuity of a Function, and Points of Discontinuity 8.1 Introduction

  The study of calculus begins with the concept of limit introduced and discussed in Chapters 7a and 7b. Of all the many consequences of this concept, one of the most important is the concept of a continuous function . One cannot think of the subject of calculus without continuous functions , which we study now.

  The word continuous means much the same in mathematics as in everyday language. We can introduce the concept of continuity proceeding from a graphic representation of a function . A function is continuous if its graph is unbroken, i.e., free from sudden jumps or gaps .

  Suppose a function is defined on an interval I . We say that the function is continuous on the interval I , if its graph consists of one continuous curve , so that it can be drawn without lifting the pencil . There is no break in any of the graphs of continuous functions (Figure 8.1a –c ).

  Figure 8.1

  If the graph of a function is broken at any point “a ” of an interval, we say that the function is not continuous (or that it is discontinuous ) at “a ”. We give the following definition:

  Definition: A function is discontinuous at x = a, if and only if it is not continuous at x = a .

  This point “a ” is called the point of discontinuity of the function. The domain of a function plays an important role in the definition of continuity (and discontinuity) of a function. A function may be continuous on one set but discontinuous on another set. It is useful to recall the definitions of the domain of definition and the natural domain

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  CounterExamples

  From Elementary Calculus to the Beginnings of Analysis

  • Andrei Bourchtein, Ludmila Bourchtein(Authors)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  2) Discontinuity of the first kind (or Jump Discontinuity). f ( x ) has a Jump Discontinuity at a , if both one-sided limits of f ( x ), as x approaches a , exist and finite, but they have different values (that is, a general limit does not exist). 3) Discontinuity of the second kind (or essential discontinuity). f ( x ) has an essential discontinuity at a , if at least one of the one-sided limits does not exist or is infinite. Remark . There are other variants (more and less detailed) of the classifi-cation of discontinuities. Local properties Remark . The properties of functions that characterize the function behav-ior in an arbitrary small neighborhood of a point are called the local properties as opposed to the global properties related to the function behavior on a chosen fixed set (in particular, on the entire domain). Continuity 55 Comparative properties 1) If f ( x ) and g ( x ) are continuous at a , and f ( x ) ≤ g ( x ) for all x ∈ X in a deleted neighborhood of a , then f ( a ) ≤ g ( a ). 2) If f ( x ) and g ( x ) are continuous at a , and f ( a ) < g ( a ), then f ( x ) < g ( x ) for all x ∈ X in a neighborhood of a . Arithmetic (algebraic) properties If f ( x ) and g ( x ) are continuous at a , then 1) f ( x ) + g ( x ) is continuous at a 2) f ( x ) − g ( x ) is continuous at a 3) f ( x ) · g ( x ) is continuous at a 4) f ( x ) g ( x ) is continuous at a (under the additional condition g ( a ) ̸ = 0) If f ( x ) is continuous at a , and α ∈ R , then 5) | f ( x ) | is continuous at a 6) ( f ( x )) α is continuous at a (under the condition α ∈ N ; or under the condition f ( a ) > 0) 7) α f ( x ) is continuous at a (under the assumption α > 0) Composite function theorem . If lim x → a f ( x ) = b , g ( x ) is continuous at b and the composite function g ( f ( x )) is defined in a deleted neighborhood of a , then lim x → a g ( f ( x )) = g ( b ).

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  eBook - PDF

  Calculus, Volume 1

  • Tom M. Apostol(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  3 CONTINUOUS FUNCTIONS 3.1 Informal description of continuity This chapter deals with the concept of continuity, one of the most important and also one of the most fascinating ideas in all of mathematics. Before we give a precise technical definition of continuity, we shall briefly discuss the concept in an informal and intuitive way to give the reader a feeling for its meaning. Roughly speaking, the situation is this: Suppose a function f has the value f ( p) at a certain point p. Then f is said to be continuous at p if at every nearby point x the function value f (x) is close to f ( p). Another way of putting it is as follows: If we let x move toward p, we want the corresponding function values f (x) to become arbitrarily close to f ( p), regardless of the manner in which x approaches p. We do not want sudden jumps in the values of a continuous function, as in the examples in Figure 3.1. (a) A Jump Discontinuity at each integer. –1 –2 –3 1 0 y x (b) An infinite discontinuity at 0. 0 y x 2 3 4 Figure 3.1 Illustrating two kinds of discontinuities. Figure 3.1(a) shows the graph of the function f defined by the equation f (x) = x − [x], where [x] denotes the greatest integer ≤ x. At each integer we have what is known as a Jump Discontinuity. For example, f (2) = 0, but as x approaches 2 from the left, f (x) approaches the value 1, which is not equal to f (2). Therefore, we have a discontinuity at 2. Note that f (x) does approach f (2) if we let x approach 2 from the right, but this by itself is not enough to establish continuity at 2. In a case like this, the function is called continuous from the right at 2 and discontinuous from the left at 2. Continuity at a point requires both continuity from the left and from the right. 126

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