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Removable Discontinuity


A real-valued univariate function f=f(x) is said to have a removable discontinuity at a point x_0 in its domain provided that both f(x_0) and

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

exist while f(x_0)!=L. Removable discontinuities are so named because one can "remove" this point of discontinuity by defining an almost everywhere identical function F=F(x) of the form

 F(x)={f(x)   for x!=x_0; L   for x=x_0,
(2)

which necessarily is everywhere-continuous.

RemovableDiscontinuity

The figure above shows the piecewise function

 f(x)={(x^2-1)/(x-1)   for x!=1; 5/2   for x=1,
(3)

a function for which lim_(x->1-)f(x)=lim_(x->1+)f(x)=2 while f(1)=5/2. In particular, f has a removable discontinuity at x=1 due to the fact that defining a function F(x) as discussed above and satisfying F(1)=2 would yield an everywhere-continuous version of f.

Note that the given definition of removable discontinuity fails to apply to functions f for which lim_(x->x_0)f(x)=L and for which f(x_0) fails to exist; in particular, the above definition allows one only to talk about a function being discontinuous at points for which it is defined. This definition isn't uniform, however, and as a result, some authors claim that, e.g., f(x)=sin(x)/x has a removable discontinuity at the point x=0. This notion is related to the so-called sinc function.

Among real-valued univariate functions, removable discontinuities are considered "less severe" than either jump or infinite discontinuities.

Unsurprisingly, one can extend the above definition in such a way as to allow the description of removable discontinuities for multivariate functions as well.

Removable discontinuities are strongly related to the notion of removable singularities.


See also

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

This entry contributed by Christopher Stover

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Cite this as:

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

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