Continuity-Removable Discontinuity
A real-valued univariate function 
is said to have a removable discontinuity at a point 
in its domain provided that both 
and
 |
(1)
|
exist while 
. Removable discontinuities are so named because one can "remove" this point of discontinuity by defining an almost everywhere identical function 
of the form
![F(x)=<span style=]() {f(x) for x!=x_0; L for x=x_0, " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/RemovableDiscontinuity/NumberedEquation2.gif" style="border:0px none; height:41px; width:158px" /> |
(2)
|
which necessarily is everywhere-continuous.
The figure above shows the piecewise function
![f(x)=<span style=]() {(x^2-1)/(x-1) for x!=1; 5/2 for x=1, " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/RemovableDiscontinuity/NumberedEquation3.gif" style="border:0px none; height:64px; width:165px" /> |
(3)
|
a function for which 
while 
. In particular, 
has a removable discontinuity at 
due to the fact that defining a function 
as discussed above 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 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., 
has a removable discontinuity at the point 
. 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
This entry contributed by Christopher Stover
