Nowhere continuous function: Difference between revisions

Revision as of 10:12, 10 March 2004

Latest revision as of 11:19, 28 June 2025

(129 intermediate revisions by 87 users not shown)

Line 1:

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

In [[mathematics]], a '''nowhere continuous''' [[function]] is a function that is not [[continuous]] at any point of its [[function ___domain|___domain]]. Suppose that ''f'' is a function from [[real number]]s to real numbers. Then, <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>. The import of this statement is 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 from the function's value.

{{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 [[function ___domain of a function|___domain]]. SupposeIf that ''<math>f''</math> is a function from [[real number]]s to real numbers. Then, then <imath>f(x)</imath> is nowhere continuous if for each point <imath>x</imath> there is ansome <imath>&epsilon;\varepsilon > 0</imath> such that for eachevery <imath>&\delta; > 0,</imath> we can find a point <imath>y</imath> such that <imath>|x - y| <&delta; \delta</imath> and <imath>|f(x) - f(y)|>&epsilon; \geq \varepsilon</imath>. TheTherefore, importno ofmatter thishow statementclose isit thatgets atto eachany fixed point, wethere canare choose aeven distance such thatcloser points arbitrarily close to our original point are taken at leastwhich thatthe distancefunction awaytakes from the function'snot-nearby valuevalues.

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 continuity definition byusing the definition of continuity in a [[topological space]].

==Examples==

One 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. If we look at this function in the vicinity of some number <i>y</i>, there are two cases:

===Dirichlet function===

If y is rational, then <i>f(y)</i>=1. To show the function is not continuous at y, we need find a single <i>&epsilon;</i> which works in the above definition. In fact, 1/2 is such an <i>&epsilon;</i>, since we can find an irrational number <i>z</i>arbitrarily close to y and <i>f(z)</i>=0, at least 1/2 away from 1. If y is irrational, then <i>f(y)</i>=0. Again, we can take <i>&epsilon;</i>=1/2, and this time we pick <i>z</i> to be an rational number as close to <i>y</i> as is required. Again, <i>f(z)</i> is more than 1/2 away from <i>f(y)</i>

{{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.

The discontinuities in this function occur because both the rational and irrational numbers are [[dense]] in the [[real number]]s. It was originally investigated by [[Johann_Peter_Gustav_Lejeune_Dirichlet|Dirichlet]].)

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]]