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{{Short description|Function which is not continuous at any point of its ___domain}}

{{refimprovemore citations needed|date=September 2012}}

In [[mathematics]], a '''nowhere continuous function''', also called an '''everywhere discontinuous function''', is a [[function (mathematics)|function]] that is not [[continuous function|continuous]] at any point of its [[___domain of a function|___domain]]. If ''<math>f''</math> is a function from [[real number]]s to real numbers, then ''<math>f''(''x'')</math> is nowhere continuous if for each point ''<math>x''</math> there is ansome <math>\varepsilon > ε&nbsp;&gt;&nbsp;0</math> such that for eachevery <math>\delta > δ&nbsp;&gt;&nbsp;0,</math> we can find a point ''<math>y''</math> such that 0< math>|''x''&nbsp;&minus;&nbsp;'' - y''|&nbsp;&lt;&nbsp;δ < \delta</math> and <math>|''f''(''x'')&nbsp;&minus;&nbsp;'' - f''(''y'')|&nbsp;≥&nbsp;ε \geq \varepsilon</math>. Therefore, no matter how close weit getgets 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]], or by using the definition of continuity in a [[topological space]].

==Examples==

==Dirichlet function==

One example of such a function is the [[indicator function]] of the [[rational number]]s, also known as the '''Dirichlet function''', named after German mathematician [[Peter Gustav Lejeune Dirichlet]].<ref>Lejeune Dirichlet, P. G. (1829) "Sur la convergence des séries trigonométriques qui servent à répresenter une fonction arbitraire entre des limites donées" [On the convergence of trigonometric series which serve to represent an arbitrary function between given limits], ''Journal für reine und angewandte Mathematik'' [Journal for pure and applied mathematics (also known as ''Crelle's Journal'')], vol. 4, pages 157 - 169.</ref> This function is written ''I''_'''Q''' and has [[___domain of a function|___domain]] and [[codomain]] both equal to the [[real number]]s. ''I''_'''Q'''(''x'') equals 1 if ''x'' is a [[rational number]] and 0 if ''x'' is not rational. If we look at this function in the vicinity of some number ''y'', there are two cases:

*If ''y'' is rational, then ''f''(''y'')&nbsp;=&nbsp;1. To show the function is not continuous at ''y'', we need to find an ε such that no matter how small we choose δ, there will be points ''z'' within δ of ''y'' such that ''f''(''z'') is not within ε of ''f''(''y'')&nbsp;=&nbsp;1. In fact, 1/2 is such an ε. Because the [[irrational number]]s are [[dense set|dense]] in the reals, no matter what δ we choose we can always find an irrational ''z'' within δ of ''y'', and ''f''(''z'')&nbsp;=&nbsp;0 is at least 1/2 away from 1.

*If ''y'' is irrational, then ''f''(''y'')&nbsp;=&nbsp;0. Again, we can take ε&nbsp;=&nbsp;1/2, and this time, because the rational numbers are dense in the reals, we can pick ''z'' to be a rational number as close to ''y'' as is required. Again, ''f''(''z'')&nbsp;=&nbsp;1 is more than 1/2 away from ''f''(''y'')&nbsp;=&nbsp;0.

In less rigorous terms, between any two irrationals, there is a rational, and vice versa.

===Dirichlet function===

The ''Dirichlet function'' can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:

{{main article|Dirichlet function}}

One example of such a function is the [[indicator function]] of the [[rational number]]s, also known as the [[Dirichlet function]]. This function is denoted as <math>\mathbf{1}_\Q</math> and has [[___domain of a function|___domain]] and [[codomain]] both equal to the [[real number]]s. By definition, <math>\mathbf{1}_\Q(x)</math> is equal to <math>1</math> if <math>x</math> is a [[rational number]] and it is <math>0</math> otherwise.

:<math>f(x)=\lim_{k\to\infty}\left(\lim_{j\to\infty}\left(\cos(k!\pi x)^{2j}\right)\right)</math>

More generally, if <math>E</math> is any subset of a [[topological space]] <math>X</math> such that both <math>E</math> and the complement of <math>E</math> are dense in <math>X,</math> then the real-valued function which takes the value <math>1</math> on <math>E</math> and <math>0</math> on the complement of <math>E</math> will be nowhere continuous. Functions of this type were originally investigated by [[Peter Gustav Lejeune Dirichlet]].<ref>{{cite journal| first = Peter Gustav | last = Lejeune Dirichlet | title = Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données| journal = Journal für die reine und angewandte Mathematik |volume = 4 | year = 1829 | url = https://eudml.org/doc/183134 | pages = 157–169}}</ref>

for integer ''j'' and ''k''.

===Non-trivial additive functions===

This shows that the ''Dirichlet function'' is a [[Baire function|Baire class]] 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a [[meagre set]].<ref>{{cite book

{{See also|Cauchy's functional equation}}

| last = Dunham

| first = William

| title = The Calculus Gallery

| publisher = Princeton University Press

| date = 2005

| pages = 197

| isbn = 0-691-09565-5 }}</ref>

A function <math>f : \Reals \to \Reals</math> is called an {{em|[[additive map|additive function]]}} if it satisfies [[Cauchy's functional equation]]:

In general, if ''E'' is any subset of a [[topological space]] ''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]].

<math display=block>f(x + y) = f(x) + f(y) \quad \text{ for all } x, y \in \Reals.</math>

For example, every map of form <math>x \mapsto c x,</math> where <math>c \in \Reals</math> is some constant, is additive (in fact, it is [[Linear map|linear]] and continuous). Furthermore, every linear map <math>L : \Reals \to \Reals</math> is of this form (by taking <math>c := L(1)</math>).

Although every [[linear map]] is additive, not all additive maps are linear. An additive map <math>f : \Reals \to \Reals</math> is linear if and only if there exists a point at which it is continuous, in which case it is continuous everywhere. Consequently, every non-linear additive function <math>\Reals \to \Reals</math> is discontinuous at every point of its ___domain.

Nevertheless, the restriction of any additive function <math>f : \Reals \to \Reals</math> to any real scalar multiple of the rational numbers <math>\Q</math> is continuous; explicitly, this means that for every real <math>r \in \Reals,</math> the restriction <math>f\big\vert_{r \Q} : r \, \Q \to \Reals</math> to the set <math>r \, \Q := \{r q : q \in \Q\}</math> is a continuous function.

Thus if <math>f : \Reals \to \Reals</math> is a non-linear additive function then for every point <math>x \in \Reals,</math> <math>f</math> is discontinuous at <math>x</math> but <math>x</math> is also contained in some [[Dense set|dense subset]] <math>D \subseteq \Reals</math> on which <math>f</math>'s restriction <math>f\vert_D : D \to \Reals</math> is continuous (specifically, take <math>D := x \, \Q</math> if <math>x \neq 0,</math> and take <math>D := \Q</math> if <math>x = 0</math>).

===Discontinuous linear maps===

{{See also|Discontinuous linear functional|Continuous linear map}}

A [[linear map]] between two [[topological vector space]]s, such as [[normed space]]s for example, is continuous (everywhere) if and only if there exists a point at which it is continuous, in which case it is even [[uniformly continuous]]. Consequently, every linear map is either continuous everywhere or else continuous nowhere.

Every [[linear functional]] is a [[linear map]] and on every infinite-dimensional normed space, there exists some [[discontinuous linear functional]].

===Other functions===

[[Conway's base 13 function]] is discontinuous at every point.

==Hyperreal characterisation==

A real function ''<math>f''</math> is nowhere continuous if its natural [[Hyperreal number|hyperreal]] extension has the property that every ''<math>x''</math> is infinitely close to a ''<math>y''</math> such that the difference ''<math>f(x) - f(y)''</math> is appreciable (i.e.that is, not [[infinitesimal]]).

==See also==

*[[Thomae%27s function]] (also known as the popcorn function) &mdash; a function that is continuous at all irrational numbers and discontinuous at all rational numbers.

* [[Blumberg theorem]]{{snd}}even if a real function <math>f : \Reals \to \Reals</math> is nowhere continuous, there is a dense subset <math>D</math> of <math>\Reals</math> such that the restriction of <math>f</math> to <math>D</math> is continuous.

* [[Thomae%27s's function]] (also known as the popcorn function) &mdash; {{snd}}a function that is continuous at all irrational numbers and discontinuous at all rational numbers.

* [[Weierstrass function]]{{snd}}a function ''continuous'' everywhere (inside its ___domain) and ''differentiable'' nowhere.

==References==

<references />

{{reflist}}

==External links==

* {{springer|title=Dirichlet-function|id=p/d032860}}

* [http://mathworld.wolfram.com/DirichletFunction.html Dirichlet Function &mdash; from MathWorld]

* [http://demonstrations.wolfram.com/TheModifiedDirichletFunction/ The 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]].

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[[Category:Mathematical analysis]]

[[Category:Topology]]

[[Category:Types of functions]]