A discontinuity of a function is a point at which the function is discontinuous.
The left figure above illustrates a discontinuity in a one-variable function, while the right figure illustrates a discontinuity
of a two-variable function
plotted as a surface in 
. In the latter case, the surface
represents the imaginary part of the natural
logarithm 
for 
in the complex numbers, with a branch
cut along the negative real axis.
Some authors refer to a discontinuity of a function as a jump, though this is rarely utilized in the literature.
For a one-variable real-valued function 
,
several common types of discontinuity are distinguished.
1. At a removable discontinuity, a finite two-sided limit exists, but the function is undefined at the point or has a different value there.
2. At a jump discontinuity, both finite one-sided limits exist but are unequal.
3. At an infinite discontinuity, at least one one-sided limit is 
or 
.
4. At an oscillatory discontinuity, the limit fails to exist because the function continues to oscillate arbitrarily near the point.
These descriptions may overlap. For a real-valued function that is bounded near an interior point, failure of the two-sided limit entails either a jump discontinuity or oscillation on at least one side. If both finite limits from the left and the right exist, they must be unequal. Otherwise, at least one side has distinct infimum limit and supremum limit.
Even with these descriptions, the sets of discontinuity of univariate real-valued functions may be stranger than expected. Indeed, functions may be discontinuous at finite sets of points, at countable sets of points which may be either isolated or dense, and on uncountable proper subsets of their domain. Other functions, such as the Dirichlet function, are discontinuous everywhere. Even so, the size of the discontinuity set of a function can say a lot about its analytic properties. For example, a theorem of Lebesgue (Royden and Fitzpatrick 2010) states that a bounded univariate real-valued function defined on a bounded interval is Riemann integrable iff it is continuous almost everywhere. Similarly, the sets of discontinuities of univariate monotone real-valued functions defined on open intervals are at most countable subsets of their domain.
With functions of two or more variables, however, no simple discontinuity classification is possible. There are a number of caveats which hinder any classification of the discontinuities of multivariate functions, chief among which is the fact that multivariate functions need neither jump nor "blow up" at points of discontinuity (Lady 1998). What's more, discontinuities of several-variable functions may occur along entire curves in the plane rather than at individual points. Various examples of discontinuous behavior are shown below.
The left function above has an infinite discontinuity at the origin. It is the function 
, which has the property
that each of the directional limits of 
tends to 
as 
approaches 
. The right function is
 |
(1)
|
a bounded function with an oscillatory discontinuity as 
. The function 
represents the surface
obtained by revolving the part of the function graph 
with 
about the 
-axis.
The two functions shown in the above figure are in some ways similar to the functions 
and 
described above. The function 
on the left has an infinite discontinuity along the line 
, with opposite infinite one-sided
limits. The function 
on the right has an
infinite discontinuity along the same line, approaching 
from both sides.
The function shown above is the piecewise function
 |
(2)
|
In particular, note that 
is monotone in each
of 
and 
separately and has jump discontinuity along
the entire line 
. This is in stark contrast to the univariate case, as
discussed above.
The function 
defined by
 |
(3)
|
is shown in the figure immediately above. Like the functions 
and 
defined previously, the function 
has a discontinuity at the point 
, though unlike those functions,
the point discontinuity of 
is more difficult to recognize. One way to observe and understand
the discontinuity of 
is by converting 
to a function of the polar
coordinates 
and 
,
the result of which is the function 
having the form
 |
(4)
|
Among other things, the expression for 
above shows that 
(and hence 
) is constant on all lines
through the origin (Lady 1998), and its dependence on
confirms that the limit
at the origin does not exist. This example is also unlike
the cases outlined for univariate functions
due to the fact that its discontinuity is essential (i.e., it is not a removable
discontinuity) and is neither a jump discontinuity
nor an infinite discontinuity.
Despite behaving very differently in terms of continuity, the sets of discontinuities of all real-valued functions in any dimension share certain properties. For example, the collection of discontinuities of such a function is always an Fsigma set; this is due to the fact that the set of all points at which a function is continuous forms a Gdelta set (Royden and Fitzpatrick 2010).
