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 ...
, a nowhere continuous function, also called an everywhere discontinuous function, is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that is not
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 ...
at any point of 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 ...
. If ''f'' is a function from
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s to real numbers, then ''f'' is nowhere continuous if for each point ''x'' there is an such that for each we can find a point ''y'' such that and . Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.
More general definitions of this kind of function can be obtained, by replacing the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
by the distance function in a
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, or by using the definition of continuity in a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
.
Dirichlet function
One example of such a function is the
indicator function of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, also known as the
Dirichlet function
In mathematics, the Dirichlet function is the indicator function 1Q or \mathbf_\Q of the set of rational numbers Q, i.e. if ''x'' is a rational number and if ''x'' is not a rational number (i.e. an irrational number).
\mathbf 1_\Q(x) = \begin
1 & ...
. This function is denoted as ''I''
Q or ''1''
Q and has
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 ...
and
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
both equal to the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. ''I''
Q(''x'') equals 1 if ''x'' is a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
and 0 if ''x'' is not rational.
More generally, if ''E'' is any subset of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
''X'' such that both ''E'' and the complement of ''E'' are dense in ''X'', then the real-valued function which takes the value 1 on ''E'' and 0 on the complement of ''E'' will be nowhere continuous. Functions of this type were originally investigated by
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
.
[ ]
Hyperreal characterisation
A real function ''f'' is nowhere continuous if its natural
hyperreal
Hyperreal may refer to:
* Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis
* Hyperreal.org, a rave culture website based in San Francisco, US
* Hyperreality, a term used in semiotics and po ...
extension has the property that every ''x'' is infinitely close to a ''y'' such that the difference is appreciable (i.e., not
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
).
See also
*
Blumberg theorem
In mathematics, the Blumberg theorem states that for any real function f : \R \to \R there is a Dense set, dense subset D of \mathbb such that the Restriction_(mathematics), restriction of f to D is continuous function, continuous.
For instance, t ...
even if a real function ''f'' : ℝ → ℝ is nowhere continuous, there is a dense subset ''D'' of ℝ such that the restriction of ''f'' to ''D'' is continuous.
*
Thomae's function
Thomae's function is a real-valued function of a real variable that can be defined as:
f(x) =
\begin
\frac &\textx = \tfrac\quad (x \text p \in \mathbb Z \text q \in \mathbb N \text\\
0 &\textx \text
\end
It is named after Carl Jo ...
(also known as the popcorn function)a function that is continuous at all irrational numbers and discontinuous at all rational numbers.
*
Weierstrass function
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
The Weierstr ...
a function ''continuous'' everywhere (inside its domain) and ''differentiable'' nowhere.
References
External links
*
Dirichlet Function — from MathWorldThe Modified Dirichlet Function{{Webarchive, url=https://web.archive.org/web/20190502165330/http://demonstrations.wolfram.com/TheModifiedDirichletFunction/ , date=2019-05-02 by George Beck,
The Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
Topology
Mathematical analysis
Types of functions