A removable discontinuity of a real-valued function 
at an accumulation point 
of its domain occurs when the
finite limit
 |
(1)
|
exists, but 
is either undefined or unequal to 
. Removable discontinuities are so named because one can "remove"
this point of discontinuity by defining an almost
everywhere identical function 
of the form
 |
(2)
|
which is continuous at 
.
The figure above shows the piecewise function
 |
(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 
.
For example, 
on its natural domain has a removable discontinuity at
,
since its limit there is 1. Extending the function
by setting 
gives the sinc function.
A removable discontinuity can be eliminated by changing the value at one point, unlike a jump discontinuity, an infinite discontinuity, or an oscillatory discontinuity.
The definition extends to removable discontinuities of multivariate functions.
Removable discontinuities are strongly related to the notion of removable singularities.
