Cauchy-continuous function

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In mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous functions have the useful property that they can always be (uniquely) extended to the Cauchy completion of their domain.

Definition

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Let   and   be metric spaces, and let   be a function from   to   Then   is Cauchy-continuous if and only if, given any Cauchy sequence   in   the sequence   is a Cauchy sequence in  

Properties

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Every uniformly continuous function is also Cauchy-continuous. Conversely, if the domain   is totally bounded, then every Cauchy-continuous function is uniformly continuous. More generally, even if   is not totally bounded, a function on   is Cauchy-continuous if and only if it is uniformly continuous on every totally bounded subset of  

Every Cauchy-continuous function is continuous. Conversely, if the domain   is complete, then every continuous function is Cauchy-continuous. More generally, even if   is not complete, as long as   is complete, then any Cauchy-continuous function from   to   can be extended to a continuous (and hence Cauchy-continuous) function defined on the Cauchy completion of   this extension is necessarily unique.

Combining these facts, if   is compact, then continuous maps, Cauchy-continuous maps, and uniformly continuous maps on   are all the same.

Examples and non-examples

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Since the real line   is complete, continuous functions on   are Cauchy-continuous. On the subspace   of rational numbers, however, matters are different. For example, define a two-valued function so that   is   when   is less than   but   when   is greater than   (Note that   is never equal to   for any rational number  ) This function is continuous on   but not Cauchy-continuous, since it cannot be extended continuously to   On the other hand, any uniformly continuous function on   must be Cauchy-continuous. For a non-uniform example on   let   be  ; this is not uniformly continuous (on all of  ), but it is Cauchy-continuous. (This example works equally well on  )

A Cauchy sequence   in   can be identified with a Cauchy-continuous function from   to   defined by   If   is complete, then this can be extended to     will be the limit of the Cauchy sequence.

Generalizations

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Cauchy continuity makes sense in situations more general than metric spaces, but then one must move from sequences to nets (or equivalently filters). The definition above applies, as long as the Cauchy sequence   is replaced with an arbitrary Cauchy net. Equivalently, a function   is Cauchy-continuous if and only if, given any Cauchy filter   on   then   is a Cauchy filter base on   This definition agrees with the above on metric spaces, but it also works for uniform spaces and, most generally, for Cauchy spaces.

Any directed set   may be made into a Cauchy space. Then given any space   the Cauchy nets in   indexed by   are the same as the Cauchy-continuous functions from   to   If   is complete, then the extension of the function to   will give the value of the limit of the net. (This generalizes the example of sequences above, where 0 is to be interpreted as  )

See also

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References

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  • Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York.