Nowhere continuous function: Difference between revisions

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

{{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 ansome {{nowrap|''ε''<math>\epsilon > 0}}</math> such that for eachevery {{nowrap|''δ''<math>\delta > 0}},</math> we can find a point ''<math>y''</math> such that {{nowrap|<math>0 < {{abs|''x'' &minus;- ''y''}}| < ''δ''}}\delta</math> and <math>|f(x) - f(y)| \geq \epsilon.</math>{{nowrap|{{abs|''f''(''x'') &minus; ''f''(''y'')}} ≥ ''ε''}}. 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]], or by using the definition of continuity in a [[topological space]].

==Examples===

==Dirichlet function==

===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 ''I''<submath>'''Q'''</sub> or ''\mathbf{1''_{'''}_\Q'''</submath> and has [[___domain of a function|___domain]] and [[codomain]] both equal to the [[real number]]s. By definition, ''I''<submath>'''\mathbf{1}_\Q'''}(''x'')</math> equalsis equal to <math>1</math> if ''<math>x''</math> is a [[rational number]] and it is <math>0</math> if ''<math>x''</math> is not rationalotherwise.

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 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>).

==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==

* [[Blumberg theorem]]{{snd}}even if a real function ''<math>f'' : ℝ\Reals →\to ℝ\Reals</math> is nowhere continuous, there is a dense subset ''<math>D''</math> of ℝ<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==

<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:Topology]]

[[Category:Mathematical analysis]]

[[Category:Topology]]

[[Category:Types of functions]]