Nowhere continuous function: Difference between revisions

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

A '''Nowhere Continuous''' [[function]] is (tautologically) a function that is not [[Continuous]] at any point. That is to say, <i>f(x)</i> is nowhere continuous for each point <i>x</i> there is an <i>&epsilon; >0</i> such that for each <i>&delta; >0</i> we can find a point <i>y</i> such that <i>|x-y|<&delta; </i> and <i>|f(x)-f(y)|>&epsilon; </i>, where the | | refers to [[absolute value]]. Basically, this is a statement that at each point we can choose a distance such that points arbitrarily close to our original point are taken at least that distance away.

{{more 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</math> is nowhere continuous if for each point <math>x</math> there is some <math>\varepsilon > 0</math> such that for every <math>\delta > 0,</math> we can find a point <math>y</math> such that <math>|x - y| < \delta</math> and <math>|f(x) - f(y)| \geq \varepsilon</math>. Therefore, no matter how close it gets 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 theby entire continuity definition byusing the definition of continuity in a [[Topologicaltopological space]].

==Examples==

On example of such a function is a function <i>f</i> on the [[real number|real numbers]] such that <i>f(x)</i> is 1 if <i>x</i> is a [[rational number]], but 0 if <i>x</i> is not rational. This satisfies the above definition with <i>&epsilon;</i> =1/2 for each <i>x</i> because both the rational and irrational numbers are [[dense]] in the [[real number|real numbers]]. This example is due to [[Johann_Peter_Gustav_Lejeune_Dirichlet|Dirichlet]]

===Dirichlet function===

{{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.

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>

===Non-trivial additive functions===

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

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

<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 (that is, not [[infinitesimal]]).

==See also==

* [[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's function]] (also known as the popcorn function){{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==

{{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]].

[[Category:Mathematical analysis]]

[[Category:Topology]]

[[Category:Types of functions]]