Classification of discontinuities

Summary

Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.

The oscillation of a function at a point quantifies these discontinuities as follows:

  • in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
  • in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits of the two sides);
  • in an essential discontinuity, oscillation measures the failure of a limit to exist; the limit is constant.

A special case is if the function diverges to infinity or minus infinity, in which case the oscillation is not defined (in the extended real numbers, this is a removable discontinuity).

Classification

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For each of the following, consider a real valued function   of a real variable   defined in a neighborhood of the point   at which   is discontinuous.

Removable discontinuity

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The function in example 1, a removable discontinuity

Consider the piecewise function

 

The point   is a removable discontinuity. For this kind of discontinuity:

The one-sided limit from the negative direction:

 
and the one-sided limit from the positive direction:
 
at   both exist, are finite, and are equal to   In other words, since the two one-sided limits exist and are equal, the limit   of   as   approaches   exists and is equal to this same value. If the actual value of   is not equal to   then   is called a removable discontinuity. This discontinuity can be removed to make   continuous at   or more precisely, the function
 
is continuous at  

The term removable discontinuity is sometimes broadened to include a removable singularity, in which the limits in both directions exist and are equal, while the function is undefined at the point  [a] This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's domain.

Jump discontinuity

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The function in example 2, a jump discontinuity

Consider the function

 

Then, the point   is a jump discontinuity.

In this case, a single limit does not exist because the one-sided limits,   and   exist and are finite, but are not equal: since,   the limit   does not exist. Then,   is called a jump discontinuity, step discontinuity, or discontinuity of the first kind. For this type of discontinuity, the function   may have any value at  

Essential discontinuity

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The function in example 3, an essential discontinuity

For an essential discontinuity, at least one of the two one-sided limits does not exist in  . (Notice that one or both one-sided limits can be  ).

Consider the function

 

Then, the point   is an essential discontinuity.

In this example, both   and   do not exist in  , thus satisfying the condition of essential discontinuity. So   is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from an essential singularity, which is often used when studying functions of complex variables).

Supposing that   is a function defined on an interval   we will denote by   the set of all discontinuities of   on   By   we will mean the set of all   such that   has a removable discontinuity at   Analogously by   we denote the set constituted by all   such that   has a jump discontinuity at   The set of all   such that   has an essential discontinuity at   will be denoted by   Of course then  

Counting discontinuities of a function

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The two following properties of the set   are relevant in the literature.

Tom Apostol[3] follows partially the classification above by considering only removable and jump discontinuities. His objective is to study the discontinuities of monotone functions, mainly to prove Froda’s theorem. With the same purpose, Walter Rudin[4] and Karl R. Stromberg[5] study also removable and jump discontinuities by using different terminologies. However, furtherly, both authors state that   is always a countable set (see[6][7]).

The term essential discontinuity has evidence of use in mathematical context as early as 1889.[8] However, the earliest use of the term alongside a mathematical definition seems to have been given in the work by John Klippert.[9] Therein, Klippert also classified essential discontinuities themselves by subdividing the set   into the three following sets:

 
 
 

Of course   Whenever     is called an essential discontinuity of first kind. Any   is said an essential discontinuity of second kind. Hence he enlarges the set   without losing its characteristic of being countable, by stating the following:

  • The set   is countable.

Rewriting Lebesgue's Theorem

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When   and   is a bounded function, it is well-known of the importance of the set   in the regard of the Riemann integrability of   In fact, Lebesgue's Theorem (also named Lebesgue-Vitali) theorem) states that   is Riemann integrable on   if and only if   is a set with Lebesgue's measure zero.

In this theorem seems that all type of discontinuities have the same weight on the obstruction that a bounded function   be Riemann integrable on   Since countable sets are sets of Lebesgue's measure zero and a countable union of sets with Lebesgue's measure zero is still a set of Lebesgue's mesure zero, we are seeing now that this is not the case. In fact, the discontinuities in the set   are absolutely neutral in the regard of the Riemann integrability of   The main discontinuities for that purpose are the essential discontinuities of first kind and consequently the Lebesgue-Vitali theorem can be rewritten as follows:

  • A bounded function,   is Riemann integrable on   if and only if the correspondent set   of all essential discontinuities of first kind of   has Lebesgue's measure zero.

The case where   correspond to the following well-known classical complementary situations of Riemann integrability of a bounded function  :

  • If   has right-hand limit at each point of   then   is Riemann integrable on   (see[10])
  • If   has left-hand limit at each point of   then   is Riemann integrable on  
  • If   is a regulated function on   then   is Riemann integrable on  

Examples

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Thomae's function is discontinuous at every non-zero rational point, but continuous at every irrational point. One easily sees that those discontinuities are all removable. By the first paragraph, there does not exist a function that is continuous at every rational point, but discontinuous at every irrational point.

The indicator function of the rationals, also known as the Dirichlet function, is discontinuous everywhere. These discontinuities are all essential of the first kind too.

Consider now the ternary Cantor set   and its indicator (or characteristic) function

 
One way to construct the Cantor set   is given by   where the sets   are obtained by recurrence according to
 

In view of the discontinuities of the function   let's assume a point  

Therefore there exists a set   used in the formulation of  , which does not contain   That is,   belongs to one of the open intervals which were removed in the construction of   This way,   has a neighbourhood with no points of   (In another way, the same conclusion follows taking into account that   is a closed set and so its complementary with respect to   is open). Therefore   only assumes the value zero in some neighbourhood of   Hence   is continuous at  

This means that the set   of all discontinuities of   on the interval   is a subset of   Since   is an uncountable set with null Lebesgue measure, also   is a null Lebesgue measure set and so in the regard of Lebesgue-Vitali theorem   is a Riemann integrable function.

More precisely one has   In fact, since   is a nonwhere dense set, if   then no neighbourhood   of   can be contained in   This way, any neighbourhood of   contains points of   and points which are not of   In terms of the function   this means that both   and   do not exist. That is,   where by   as before, we denote the set of all essential discontinuities of first kind of the function   Clearly  

Discontinuities of derivatives

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Let now   an open interval and  the derivative of a function,  , differentiable on  . That is,   for every  .

It is well-known that according to Darboux's Theorem the derivative function   has the restriction of satisfying the intermediate value property.

  can of course be continuous on the interval  . Recall that any continuous function, by Bolzano's Theorem, satisfies the intermediate value property.

On the other hand, the intermediate value property does not prevent   from having discontinuities on the interval  . But Darboux's Theorem has an immediate consequence on the type of discontinuities that   can have. In fact, if   is a point of discontinuity of  , then necessarily   is an essential discontinuity of  .[11]

This means in particular that the following two situations cannot occur:

  1.   is a removable discontinuity of  .
  2.   is a jump discontinuity of  .

Furtherly, two other situations have to be excluded (see John Klippert[12]):

  1.  
  2.  

Observe that whenever one of the conditions (i), (ii), (iii), or (iv) is fulfilled for some   one can conclude that   fails to possess an antiderivative,  , on the interval  .

On the other hand, a new type of discontinuity with respect to any function   can be introduced: an essential discontinuity,  , of the function  , is said to be a fundamental essential discontinuity of   if

 
and
 

Therefore if   is a discontinuity of a derivative function  , then necessarily   is a fundamental essential discontinuity of  .

Notice also that when   and   is a bounded function, as in the assumptions of Lebesgue's Theorem, we have for all  :

 
 
and
 
Therefore any essential discontinuity of   is a fundamental one.

See also

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  • Removable singularity – Undefined point on a holomorphic function which can be made regular
  • Mathematical singularity – Point where a function, a curve or another mathematical object does not behave regularly
  • Extension by continuity – topological space in which a point and a closed set are, if disjoint, separable by neighborhoods
  • Smoothness – Number of derivatives of a function (mathematics)
    • Geometric continuity – Number of derivatives of a function (mathematics)
    • Parametric continuity – Number of derivatives of a function (mathematics)

Notes

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  1. ^ See, for example, the last sentence in the definition given at Mathwords.[1]

References

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  1. ^ "Mathwords: Removable Discontinuity".
  2. ^ Stromberg, Karl R. (2015). An Introduction to Classical Real Analysis. American Mathematical Society. pp. 120. Ex. 3 (c). ISBN 978-1-4704-2544-9.
  3. ^ Apostol, Tom (1974). Mathematical Analysis (second ed.). Addison and Wesley. pp. 92, sec. 4.22, sec. 4.23 and Ex. 4.63. ISBN 0-201-00288-4.
  4. ^ Walter, Rudin (1976). Principles of Mathematical Analysis (third ed.). McGraw-Hill. pp. 94, Def. 4.26, Thms. 4.29 and 4.30. ISBN 0-07-085613-3.
  5. ^ Stromberg, Karl R. Op. cit. pp. 128, Def. 3.87, Thm. 3.90.
  6. ^ Walter, Rudin. Op. cit. pp. 100, Ex. 17.
  7. ^ Stromberg, Karl R. Op. cit. pp. 131, Ex. 3.
  8. ^ Whitney, William Dwight (1889). The Century Dictionary: An Encyclopedic Lexicon of the English Language. Vol. 2. London and New York: T. Fisher Unwin and The Century Company. p. 1652. ISBN 9781334153952. Archived from the original on 2008-12-16. An essential discontinuity is a discontinuity in which the value of the function becomes entirely indeterminable.
  9. ^ Klippert, John (February 1989). "Advanced Advanced Calculus: Counting the Discontinuities of a Real-Valued Function with Interval Domain". Mathematics Magazine. 62: 43–48. doi:10.1080/0025570X.1989.11977410 – via JSTOR.
  10. ^ Metzler, R. C. (1971). "On Riemann Integrability". American Mathematical Monthly. 78 (10): 1129–1131. doi:10.1080/00029890.1971.11992961.
  11. ^ Rudin, Walter. Op.cit. pp. 109, Corollary.
  12. ^ Klippert, John (2000). "On a discontinuity of a derivative". International Journal of Mathematical Education in Science and Technology. 31:S2: 282–287. doi:10.1080/00207390050032252.

Sources

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  • Malik, S.C.; Arora, Savita (1992). Mathematical Analysis (2nd ed.). New York: Wiley. ISBN 0-470-21858-4.
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