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Jump Discontinuity


A real-valued univariate function f=f(x) has a jump discontinuity at a point x_0 in its domain provided that

 lim_(x->x_0-)f(x)=L_1<infty
(1)

and

 lim_(x->x_0+)f(x)=L_2<infty
(2)

both exist and that L_1!=L_2.

The notion of jump discontinuity shouldn't be confused with the rarely-utilized convention whereby the term jump is used to define any sort of functional discontinuity.

JumpDiscontinuity

The figure above shows an example of a function having a jump discontinuity at a point in its domain.

Though less algebraically-trivial than removable discontinuities, jump discontinuities are far less ill-behaved than other types of singularities such as infinite discontinuities. This fact can be seen in a number of scenarios, e.g., in the fact that univariate monotone functions can have at most countably many discontinuities (Royden and Fitzpatrick 2010), the worst of which can be jump discontinuities (Zakon 2004).

JumpDiscontinuityMonotoneLine

Unsurprisingly, the definition given above can be generalized to include jump discontinuities in multivariate real-valued functions as well. For example, the function shown in this figure is the piecewise function

 t(x,y)={(x,x)   for x+y>1; (-5+x,-5+x)   for x+y<=1,
(3)

a function which is monotone in each of x and y separately and has jump discontinuity along the entire line x+y=1.


See also

Branch Cut, Continuous, Discontinuity, Discontinuous, Discontinuous Function, Essential Singularity, Infinite Discontinuity, Isolated Singularity, Polar Coordinates, Pole, Removable Discontinuity, Removable Singularity, Singular Point, Singularity

This entry contributed by Christopher Stover

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References

Royden, H. L. and Fitzpatrick, P. M. Real Analysis. Pearson, 2010.Zakon, E. Mathematical Analysis Volume 1. West Lafayette, IN: The Trilla Group, 2004. http://www.trillia.com/zakon-analysisI.html.

Cite this as:

Stover, Christopher. "Jump Discontinuity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/JumpDiscontinuity.html

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