# Cauchy-continuous function

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In [mathematics](/source/mathematics), a '''Cauchy-continuous''', or '''Cauchy-regular''', function is a special kind of [continuous function](/source/continuous_function) between [metric space](/source/metric_space)s (or more general spaces). Cauchy-continuous functions have the useful property that they can always be (uniquely) extended to the [Cauchy completion](/source/Cauchy_completion) of their domain.

== Definition ==

Let <math>X</math> and <math>Y</math> be [metric space](/source/metric_space)s, and let <math>f : X \to Y</math> be a [function](/source/Function_(mathematics)) from <math>X</math> to <math>Y.</math> Then <math>f</math> is Cauchy-continuous if and only if, given any [Cauchy sequence](/source/Cauchy_sequence) <math>\left(x_1, x_2, \ldots\right)</math> in <math>X,</math> the sequence <math>\left(f\left(x_1\right), f\left(x_2\right), \ldots\right)</math> is a Cauchy sequence in <math>Y.</math>

== Properties ==

Every [uniformly continuous function](/source/uniformly_continuous_function) is also Cauchy-continuous. Conversely, if the domain <math>X</math> is [totally bounded](/source/Totally_bounded_space), then every Cauchy-continuous function is uniformly continuous. More generally, even if <math>X</math> is not totally bounded, a function on <math>X</math> is Cauchy-continuous if and only if it is uniformly continuous on every totally bounded subset of <math>X.</math>

Every Cauchy-continuous function is [continuous](/source/Continuous_function). Conversely, if the domain <math>X</math> is [complete](/source/Complete_space), then every continuous function is Cauchy-continuous. More generally, even if <math>X</math> is not complete, as long as <math>Y</math> is complete, then any Cauchy-continuous function from <math>X</math> to <math>Y</math> can be extended to a continuous (and hence Cauchy-continuous) function defined on the [Cauchy completion](/source/Cauchy_completion) of <math>X;</math> this extension is necessarily unique.

Combining these facts, if <math>X</math> is [compact](/source/Compact_metric_space), then continuous maps, Cauchy-continuous maps, and uniformly continuous maps on <math>X</math> are all the same.

== Examples and non-examples ==

Since the [real line](/source/real_line) <math>\R</math> is complete, continuous functions on <math>\R</math> are Cauchy-continuous. On the [subspace](/source/Subspace_(topology)) <math>\Q</math> of [rational number](/source/rational_number)s, however, matters are different. For example, define a two-valued function so that <math>f(x)</math> is <math>0</math> when <math>x^2</math> is less than <math>2</math> but <math>1</math> when <math>x^2</math> is greater than <math>2.</math> (Note that <math>x^2</math> is never equal to <math>2</math> for any rational number <math>x.</math>) This function is continuous on <math>\Q</math> but not Cauchy-continuous, since it cannot be extended continuously to <math>\R.</math> On the other hand, any uniformly continuous function on <math>\Q</math> must be Cauchy-continuous. For a non-uniform example on <math>\Q,</math> let <math>f(x)</math> be <math>2^x</math>; this is not uniformly continuous (on all of <math>\Q</math>), but it is Cauchy-continuous. (This example works equally well on <math>\R.</math>)

A Cauchy sequence <math>\left(y_1, y_2, \ldots\right)</math> in <math>Y</math> can be identified with a Cauchy-continuous function from <math>\left\{1, 1/2, 1/3, \ldots\right\}</math> to <math>Y,</math> defined by <math>f\left(1/n\right) = y_n.</math> If <math>Y</math> is complete, then this can be extended to <math>\left\{1, 1/2, 1/3, \ldots\right\}\cup\{0\};</math> <math>f(0)</math> will be the limit of the Cauchy sequence.

== Generalizations ==

Cauchy continuity makes sense in situations more general than metric spaces, but then one must move from sequences to [net](/source/Net_(topology))s (or equivalently [filter](/source/Filter_(topology))s). The definition above applies, as long as the Cauchy sequence <math>\left(x_1, x_2, \ldots\right)</math> is replaced with an arbitrary [Cauchy net](/source/Cauchy_net). Equivalently, a function <math>f</math> is Cauchy-continuous if and only if, given any [Cauchy filter](/source/Cauchy_filter) <math>\mathcal{F}</math> on <math>X,</math> then <math>f(\mathcal{F})</math> is a Cauchy filter base on <math>Y.</math> This definition agrees with the above on metric spaces, but it also works for [uniform space](/source/uniform_space)s and, most generally, for [Cauchy space](/source/Cauchy_space)s.

Any [directed set](/source/directed_set) <math>A</math> may be made into a Cauchy space. Then given any space <math>Y,</math> the Cauchy nets in <math>Y</math> indexed by <math>A</math> are the same as the Cauchy-continuous functions from <math>A</math> to <math>Y.</math> If <math>Y</math> is complete, then the extension of the function to <math>A \cup \{\infty\}</math> will give the value of the limit of the net. (This generalizes the example of sequences above, where 0 is to be interpreted as <math>\frac{1}{\infty}.</math>)

== See also ==

* {{annotated link|Cauchy space}}
* [Heine–Cantor theorem](/source/Heine%E2%80%93Cantor_theorem)

== References ==

* Eva Lowen-Colebunders (1989). <cite>Function Classes of Cauchy Continuous Maps</cite>. Dekker, New York.

Category:Topology
Category:Types of functions

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Adapted from the Wikipedia article [Cauchy-continuous function](https://en.wikipedia.org/wiki/Cauchy-continuous_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Cauchy-continuous_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.