In
mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a
function that is not
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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 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 sett ...
, 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 po ...
.
Dirichlet function
One example of such a function is the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x ...
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 ra ...
s, also known as the
Dirichlet function. 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 ...
both equal to the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 ra ...
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 po ...
''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 extension has the property that every ''x'' is infinitely close to a ''y'' such that the difference is appreciable (i.e., not
infinitesimal).
See also
*
Blumberg theoremeven 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 (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 Weierst ...
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.
Topology
Mathematical analysis
Types of functions