A real-valued univariate function 
is said to have an infinite discontinuity at a point 
in its domain provided that either
(or both) of the lower or upper
limits of 
fails to exist as 
tends to 
.
Infinite discontinuities are sometimes referred to as essential discontinuities, phraseology indicative of the fact that such points of discontinuity are considered to be "more severe" than either removable or jump discontinuities.
The figure above shows the piecewise function
 |
(1)
|
a function for which both 
and 
fail to exist. In particular, 
has an infinite discontinuity at 
.
It is not uncommon for authors to say that univariate functions 
defined on a domain 
and admitting vertical asymptotes of the form 
have infinite discontinuities there though, strictly speaking,
this terminology is incorrect unless such functions are defined piecewise so that
.
For example, the function 
has vertical asymptotes at 
, 
, though it has no discontinuities of any kind on its
domain.
Unsurprisingly, one can extend the above definition to infinite discontinuities of multivariate functions as well.
