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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.