Continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s are of utmost importance in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, functions and applications. However, not all
functions are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
. If a function is not continuous at a point in its
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
, one says that it has a discontinuity there. The
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of all points of discontinuity of a function may be a
discrete set
]
In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...
, a
dense set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the r ...
, or even the entire domain of the function. This article describes the classification of discontinuities in the simplest case of functions of a single
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
taking real values.
The
oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
of a function at a point quantifies these discontinuities as follows:
* in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
* in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits of the two sides);
* in an essential
discontinuity, oscillation measures the failure of a
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
to exist; the limit is constant.
A special case is if the function diverges to
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
or minus
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
, in which case the
oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
is not defined (in the extended real numbers, this is a removable discontinuity).
Classification
For each of the following, consider a
real valued function of a real variable
defined in a neighborhood of the point
at which
is discontinuous.
Removable discontinuity
![Discontinuity removable](https://upload.wikimedia.org/wikipedia/commons/9/9d/Discontinuity_removable.eps.png)
Consider the
piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
function
The point
is a ''removable
discontinuity''. For this kind of discontinuity:
The
one-sided limit
In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right.
The limit as x decreases in value approaching a (x approaches ...
from the negative direction:
and the one-sided limit from the positive direction:
at
''both'' exist, are finite, and are equal to
In other words, since the two one-sided limits exist and are equal, the limit
of
as
approaches
exists and is equal to this same value. If the actual value of
is ''not'' equal to
then
is called a . This discontinuity can be removed to make
continuous at
or more precisely, the function
is continuous at
The term ''removable discontinuity'' is sometimes broadened to include a
removable singularity
In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourh ...
, in which the limits in both directions exist and are equal, while the function is
undefined
Undefined may refer to:
Mathematics
* Undefined (mathematics), with several related meanings
** Indeterminate form, in calculus
Computing
* Undefined behavior, computer code whose behavior is not specified under certain conditions
* Undefined ...
at the point
This use is an
abuse of terminology
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors an ...
because
continuity and discontinuity of a function are concepts defined only for points in the function's domain.
Jump discontinuity
![Discontinuity jump](https://upload.wikimedia.org/wikipedia/commons/e/e6/Discontinuity_jump.eps.png)
Consider the function
Then, the point
is a '.
In this case, a single limit does not exist because the one-sided limits,
and
exist and are finite, but are not equal: since,
the limit
does not exist. Then,
is called a ''jump discontinuity'', ''step discontinuity'', or ''discontinuity of the first kind''. For this type of discontinuity, the function
may have any value at
Essential discontinuity
![Discontinuity essential](https://upload.wikimedia.org/wikipedia/commons/9/92/Discontinuity_essential.svg)
For an essential discontinuity, at least one of the two one-sided limits does not exist in
. (Notice that one or both one-sided limits can be
).
Consider the function
Then, the point
is an '.
In this example, both
and
do not exist in
, thus satisfying the condition of essential discontinuity. So
is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from an
essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.
The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
, which is often used when studying
functions of complex variables).
Supposing that
is a function defined on an interval
we will denote by
the set of all discontinuities of
on
By
we will mean the set of all
such that
has a ''removable'' discontinuity at
Analogously by
we denote the set constituted by all
such that
has a ''jump'' discontinuity at
The set of all
such that
has an ''essential'' discontinuity at
will be denoted by
Of course then
Counting discontinuities of a function
The two following properties of the set
are relevant in the literature.
* The set of
is an
set. The set of points at which a function is continuous is always a
set (see).
* If on the interval
is monotone then
is
at most countable and
This is
Froda's theorem
In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and th ...
.
Tom Apostol follows partially the classification above by considering only removal and jump discontinuities. His objective is to study the discontinuities of monotone functions, mainly to prove Froda’s theorem. With the same purpose, Walter Rudin and Karl R. Stromberg study also removal and jump discontinuities by using different terminologies. However, furtherly, both authors state that
is always a countable set (see).
The term essential discontinuity seems to have been introduced by John Klippert. Furtherly he also classified essential discontinuities themselves by subdividing the set
into the three following sets:
Of course
Whenever
is called an ''essential discontinuity of first kind''. Any
is said an ''essential discontinuity of second kind.'' Hence he enlarges the set
without losing its characteristic of being countable, by stating the following:
* The set
is countable.
Rewriting Lebesgue's Theorem
When