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| {{redirect|Jump point|the science-fiction concept|jump drive}}
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| {{Refimprove|date=March 2013}}
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| [[Continuous function]]s are of utmost importance in [[mathematics]], functions and applications. However, not all [[function (mathematics)|functions]] are continuous. If a function is not continuous at a point in its [[domain (mathematics)|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.
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| This article describes the '''classification of discontinuities''' in the simplest case of functions of a single [[real number|real]] variable taking real values.
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| Consider a real valued function ''ƒ'' of a real variable ''x'', defined in a neighborhood of the point ''x''<sub>0</sub> at which ''ƒ'' is discontinuous. Three situations can be distinguished:
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| {{ordered list
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| |1= The [[one-sided limit]] from the negative direction
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| :<math>L^{-}=\lim_{x\to x_0^{-}} f(x)</math>
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| and the one-sided limit from the positive direction | |
| :<math>L^{+}=\lim_{x\to x_0^{+}} f(x)</math>
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| at <math>\scriptstyle x_0</math> exist, are finite, and are equal to <math>\scriptstyle L \;=\; L^{-} \;=\; L^{+}</math>. Then, if ''ƒ''(''x''<sub>0</sub>) is not equal to <math>\scriptstyle L</math>, ''x''<sub>0</sub> is called a ''removable discontinuity''. This discontinuity can be 'removed to make ''ƒ'' continuous at ''x''<sub>0</sub>', or more precisely, the function
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| :<math>g(x) = \begin{cases}f(x) & x\ne x_0 \\ L & x = x_0\end{cases}</math> | |
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| is continuous at ''x''=''x''<sub>0</sub>.
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| |2= The limits <math>\scriptstyle L^{-}</math> and <math>\scriptstyle L^{+}</math> exist and are finite, but not equal. Then, ''x''<sub>0</sub> is called a ''jump discontinuity'' or ''step discontinuity''. For this type of discontinuity, the function ''ƒ'' may have any value at ''x''<sub>0</sub>.
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| |3= One or both of the limits <math>\scriptstyle L^{-}</math> and <math>\scriptstyle L^{+}</math> does not exist or is infinite. Then, ''x''<sub>0</sub> is called an ''essential discontinuity'', or ''infinite discontinuity''. (This is distinct from the term ''[[essential singularity]]'' which is often used when studying [[complex analysis|functions of complex variables]].)
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| }}
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| The term ''removable discontinuity'' is sometimes used by [[abuse of terminology]] for cases in which the limits in both directions exist and are equal, while the function is [[defined and undefined|undefined]] at the point <math>\scriptstyle x_0</math>.<ref>See, for example, the last sentence in the definition given at Mathwords.[http://www.mathwords.com/r/removable_discontinuity.htm]</ref> This use is abusive because [[Continuous function|continuity]] and discontinuity of a function are concepts defined only for points in the function's domain. Such a point not in the domain is properly named a [[removable singularity]].
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| The [[Oscillation (mathematics)|oscillation]] of a function at a point quantifies these discontinuities as follows:
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| * in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
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| * in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits from the two sides);
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| * in an essential discontinuity, oscillation measures the failure of a limit to exist.
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| ==Examples==
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| [[File:Discontinuity removable.eps.png|thumb|left|The function in example 1, a removable discontinuity]]
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| 1. Consider the function
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| :<math>f(x) = \begin{cases}
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| x^2 & \mbox{ for } x < 1 \\
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| 0 & \mbox{ for } x = 1 \\
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| 2-x & \mbox{ for } x > 1
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| \end{cases}</math>
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| Then, the point <math>\scriptstyle x_0 \;=\; 1</math> is a ''removable discontinuity''.
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| {{clear}}
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| [[File:Discontinuity jump.eps.png|thumb|left|The function in example 2, a jump discontinuity]]
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| 2. Consider the function
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| :<math>f(x) = \begin{cases}
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| x^2 & \mbox{ for } x < 1 \\
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| 0 & \mbox{ for } x = 1 \\
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| 2 - (x-1)^2 & \mbox{ for } x > 1
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| \end{cases}</math>
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| Then, the point <math>\scriptstyle x_0 \;=\; 1</math> is a ''jump discontinuity''.
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| {{clear}}
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| [[File:Discontinuity essential.eps.png|thumb|left|The function in example 3, an essential discontinuity]]
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| 3. Consider the function
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| :<math>f(x) = \begin{cases}
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| \sin\frac{5}{x-1} & \mbox{ for } x < 1 \\
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| 0 & \mbox{ for } x = 1 \\
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| \frac{1}{x-1} & \mbox{ for } x > 1
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| \end{cases}</math>
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| Then, the point <math>\scriptstyle x_0 \;=\; 1</math> is an ''essential discontinuity (sometimes called infinite discontinuity)''. For it to be an essential discontinuity, it would have sufficed that only one of the two one-sided limits did not exist or were infinite. However, given this example the discontinuity is also an ''essential discontinuity'' for the extension of the function into complex variables.
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| {{clear}}
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| ==The set of discontinuities of a function==
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| The set of points at which a function is continuous is always a [[G-delta set|G<sub>δ</sub> set]]. The set of discontinuities is an [[F-sigma set|F<sub>σ</sub> set]].
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| The set of discontinuities of a [[monotonic function]] is [[countable|at most countable]]. This is [[Froda's theorem]].
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| ''[[Thomae's function]]'' is discontinuous at every [[rational point]], but continuous at every irrational point.
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| The [[indicator function]] of the rationals, also known as the ''[[Dirichlet function]]'', is [[nowhere continuous|discontinuous everywhere]].
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| ==See also==
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| *[[Removable singularity]]
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| *[[Mathematical singularity]]
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| *[[Regular_space#Extension_by_continuity|Extension by continuity]]
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| ==Notes==
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| <references />
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| ==References==
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| *{{cite book
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| | last = Malik
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| | first = S. C.
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| | coauthors = Arora, Savita
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| | title = Mathematical analysis, 2nd ed
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| | publisher = New York: Wiley
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| | year = 1992
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| | pages =
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| | isbn = 0-470-21858-4
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| }}
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| ==External links==
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| * {{planetmath reference|title=Discontinuous|id=4447}}
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| * [http://demonstrations.wolfram.com/Discontinuity/ "Discontinuity"] by [[Ed Pegg, Jr.]], [[The Wolfram Demonstrations Project]], 2007.
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| * {{MathWorld | urlname=Discontinuity | title=Discontinuity}}
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| * {{SpringerEOM| title=Discontinuity point | id=Discontinuity_point | oldid=12112 | first=L.D. | last=Kudryavtsev }}
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| [[Category:Mathematical analysis]]
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