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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 {{nowrap|''ε''<math>\varepsilon > 0}}</math> such that for eachevery {{nowrap|''δ''<math>\delta > 0}},</math> we can find a point ''<math>y''</math> such that {{nowrap|0 < {{absmath>|''x'' &minus;- ''y''}}| < ''δ''}}\delta</math> and {{nowrap|{{abs<math>|''f''(''x'') &minus;- ''f''(''y'')}}| ≥\geq ''ε''}}\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 {{nowrap|1=''f''(''y'') = 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 {{nowrap|1=''f''(''y'') = 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 {{nowrap|1=''f''(''z'') = 0}} is at least 1/2 away from 1.

*If ''y'' is irrational, then {{nowrap|1=''f''(''y'') = 0}}. Again, we can take {{nowrap|1=''ε'' = 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, {{nowrap|1=''f''(''z'') = 1}} is more than 1/2 away from {{nowrap|1=''f''(''y'') = 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)\right)^{2j}\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 {{nowrap|''<math>f''(''x'') −- ''f''(''y'')}}</math> is appreciable (i.e.that is, not [[infinitesimal]]).

==See also==

*[[Thomae's 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.

*[[Weierstrass function]]:A function 'Continuous' everywhere(inside ___domain) and 'Differentiable' nowhere.

* [[Thomae'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]]:A{{snd}}a function 'Continuous'continuous'' everywhere (inside its ___domain) and 'Differentiable'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]]