Removable Discontinuity

3 Key excerpts on "Removable 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)

  b ). Such a discontinuity is called Removable Discontinuity, for obvious reasons.

  It is not always possible to remove the discontinuity of a function. If the discontinuity is not removable it is called an irremovable (or an essential) discontinuity of the function, as mentioned earlier. If does not exist then f (x ) is said to have an irremovable (or essential) discontinuity at x = a . (Note that the graphs of the functions in Figures 8.2a and 8.3 indicate the point of removable discontinuities whereas those displayed in Figures 8.2b , 8.4 , 8.5 , and 8.6 indicate the points of irremovable discontinuities .)

  The simplest type of essential discontinuity occurs at those points at which a function makes a (finite) jump, that is, where the function has a definite limit as x → a − and a different definite limit as x → a + . Such discontinuities are displayed in Figures 8.4 and 8.5 .

  Note (7): It must be clear that if the graph of the function has a finite jump of a point alone, then the function is said to have Removable Discontinuity at that point. But, if there is a finite jump of a portion of the curve , then such a function has irremovable (or essential) discontinuity at the point of jump.

  Remark: In the case of an irRemovable Discontinuity it does not matter whether or how the function is defined at the point of discontinuity. This will be clear from the following example, and many more later on.

  Example (5): Recall the function f (x ) = 1/x , x ≠ 0. Clearly this function is not continuous at x = 0, and in any interval containing the point “0”. The examination of the graph of 1/x in the vicinity of the point x = 0 clearly shows that it splits into two separate curves at the point x = 0 (see Figure 8.9a ). Further note that, in this case, we cannot make “f ” continuous by assigning any value to f (0). Also observe that neither exists nor f (0) is defined. We say that “f ” has an infinite discontinuity at x = 0 . This is an essential discontinuity

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  eBook - PDF

  Introduction to Analysis

  • Corey M. Dunn(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  Such a → → discontinuity is called a discontinuity of the first kind . c c Jump discontinuity at x = c Removable Discontinuity at x = c FIGURE 5.4.2 Discontinuity of the first kind. Other discontinuities, pictured in Figure 5.4.3, are not so nice. A discon-tinuity not of the first kind is called a discontinuity of the second kind . Definition 5.40. Suppose f : D R is discontinuous at c ∈ D . → 285 Limits and Continuity c c FIGURE 5.4.3 Discontinuities of the second kind. 1. A discontinuity of the first kind is one where lim x c + f ( x ) and → lim x c − f ( x ) exist. → 2. A Removable Discontinuity is a discontinuity of the first kind where lim x c + f ( x ) = lim x c − f ( x ). → → 3. A discontinuity of the second kind is one where either lim x c + f ( x ) or lim x c − f ( x ) does not exist (in R ). → → The term “Removable Discontinuity” makes sense: if f has a Removable Discontinuity at c , then there is a (unique) way to assign a value of f ( c ) so that the resulting function is continuous. We have seen this already in the x discussion (and subsequent practice problem) on Page 268. If f ( x ) = x 2 − − 1 1 , then f is continuous on R − { 1 } . 1 2 f ( x ) FIGURE 5.4.4 The graph of f ( x ) = x x 2 − − 1 1 . 286 Introduction to Analysis Since 2 = lim x 2 − 1 = lim x 2 − 1 = lim x 2 − 1 , x 1 x 1 + x 1 − → x − 1 → x − 1 → x − 1 if we were to define f (1) to be anything other than 2, this function would have a Removable Discontinuity at x = 1 (see Figure 5.4.4 ). But defining f (1) = 2 would “remove” this discontinuity, and produce a continuous function on all of R : � x 2 − 1 for x = 1 , f ( x ) extends to x + 1 = x − 1 � 2 for x = 1 . The study of discontinuities is a very interesting one throughout analysis. One basic realm is in the study of discontinuities of monotone functions.

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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)

  Since the behavior of a function at an isolated point is hardly of any interest, 53 54 Counterexamples: From Calculus to the Beginnings of Analysis we will usually consider the definition of continuity by limit as a complete definition. One-sided continuity . A function f ( x ) defined on X is right-hand (left-hand) continuous at a point a ∈ X if for every ε > 0 there exists δ > 0 such that whenever x ∈ X and 0 ≤ x − a < δ ( − δ < x − a ≤ 0) it follows that | f ( x ) − f ( a ) | < ε . Continuity on a set . A function f ( x ) defined on X is continuous on a set S ⊂ X , if f ( x ) is continuous at every point of S . Remark . It is worth to notice that in the case when endpoints belong to the interval, for example a closed interval [ a, b ], continuity on [ a, b ] means continuity at every point c ∈ ( a, b ) plus right-hand continuity at a and left-hand continuity at b . Discontinuity point . Let a be a limit point of the domain of f ( x ). If f ( x ) is not continuous at a , then a is a point of discontinuity of f ( x ) (or equivalently, f ( x ) has a discontinuity at a ). Remark . Sometimes it is required that a should be a point of the domain of f ( x ). We will not impose this restriction. Classification of discontinuities . 1) Removable Discontinuity. f ( x ) has a Removable Discontinuity at a , if the finite limit of f ( x ), as x approaches a , exists, but the function is not defined at a or has the value different from the value of the limit. It is called removable, because the proper redefinition of the original function just at the point a removes this discontinuity. 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).

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