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 some <math>\epsilonvarepsilon > 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 \epsilon.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]].

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{{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> if <math>x</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>

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{{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>).

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===Other functions===

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

==Hyperreal characterisation==