Classification of discontinuities
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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. This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values.
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.
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.
Consider the piecewise function
The point is a removable discontinuity. For this kind of discontinuity:
The one-sided limit from the negative direction:
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.
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
For an essential discontinuity, at least one of the two one-sided limits doesn't exist. Consider the function
Then, the point is an essential discontinuity.
In this example, both and does not exist, 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 that
Counting discontinuities of a function
The two following properties of the set are relevant in the literature.
- The set of is an set. The set of points at which a function is continuous is always a set (see ).
Tom Apostol follows partially the classification above by considering only removal 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 and Karl R. Stromberg study also removal and jump discontinuities by using different terminologies. However, furtherly, both authors state that is always a countable set (see).
The term essential discontinuity seems to have been introduced by John Klippert where essential discontinuities are themselves classified by subdividing the set into the three following sets:
Of course Whenever is called an essential discontinuity of first kind. For any it will be said an essential discontinuity of second kind. Hence the following important property is proved:
- The set is countable.
Rewriting Lebesgue's Theorem
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 absolutly 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 )
- 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
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 essential of the first kind, that is By the first paragraph, there does not exist a function that is continuous at every rational point, but discontinuous at every irrational point.
Consider now the ternary Cantor set and its indicator (or characteristic) function
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 a noncountable 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 rare (closed of empty interior) 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
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 doesn't 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 .
This means in particular that the following two situations cannot occur:
- is a removable discontinuity of .
- is a jump discontinuity of .
Furtherly, two other situations have to be excluded (see John Klippert):
Observe that whenever one of the conditions , , or 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
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 is an essential discontinuity of first kind of if and only if is a fundamental essential discontinuity of . Consequently the set considered in that theorem is the set of all fundamental essential discontinuities.
- Removable singularity
- Mathematical singularity
- Extension by continuity
- Smoothness – Property measuring how many times a function can be differentiated
- See, for example, the last sentence in the definition given at Mathwords.
- "Mathwords: Removable Discontinuity".
- Stromberg, Karl R. (2015). An Introduction to Classical Real Analysis. American Mathematical Society. pp. 120. Ex. 3 (c). ISBN 978-1-4704-2544-9.
- 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.
- 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.
- Stromberg, Karl R. Op. cit. pp. 128, Def. 3.87, Thm. 3.90.
- Walter, Rudin. Op. cit. pp. 100, Ex. 17.
- Stromberg, Karl R. Op. cit. pp. 131, Ex. 3.
- 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.
- Metzler, R. C. (1971). "On Riemann Integrability". American Mathematical Monthly. 78 (10): 1129–1131. doi:10.1080/00029890.1971.11992961.
- Rudin, Walter. Op.cit. pp. 109, Corollary.
- Klippert, John (2000). "On a discontinuity of a derivative". International Journal of Mathematical Education in Science and Technology. 31:S2: 282–287.
- Malik, S.C.; Arora, Savita (1992). Mathematical Analysis (2nd ed.). New York: Wiley. ISBN 0-470-21858-4.