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
 |
(2)
|
which necessarily is everywhere-continuous.
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 
.
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.
