What Is A Removable Discontinuity

Removable Discontinuity

A real-valued univariate function is said to have a removable aperture at a point in its domain provided that both and

(ane)

be while . Removable discontinuities are so named because one can "remove" this signal of discontinuity by defining an almost everywhere identical function of the course

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

(2)

which necessarily is everywhere-continuous.

RemovableDiscontinuity

The figure above shows the piecewise function

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

(three)

a role for which while . In item, has a removable discontinuity at due to the fact that defining a role equally discussed to a higher place and satisfying would yield an everywhere-continuous version of .

Note that the given definition of removable discontinuity fails to apply to functions for which and for which fails to exist; in particular, the to a higher place definition allows ane only to talk about a part being discontinuous at points for which it is defined. This definition isn't uniform, still, and every bit a upshot, some authors claim that, due east.g., has a removable discontinuity at the point . This notion is related to the so-called sinc function.

Amidst existent-valued univariate functions, removable discontinuities are considered "less severe" than either spring or infinite discontinuities.

Unsurprisingly, i can extend the in a higher place 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 as well

Co-operative Cutting, Continuous, Aperture, Discontinuous, Discontinuous Role, Essential Singularity, Infinite Discontinuity, Isolated Singularity, Leap Aperture, Polar Coordinates, Pole, Removable Singularity, Atypical Indicate, Singularity

This entry contributed by Christopher Stover

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Cite this every bit:

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