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

be metric spaces be a function is Cauchy-continuous if and only if, given any Cauchy sequence

Properties

Every uniformly continuous function is totally bounded

Every Cauchy-continuous function is continuous is complete can be extended to a continuous (and hence Cauchy-continuous) function defined on the Cauchy completion this extension is necessarily unique.

is compact are all the same.

Examples and non-examples

Since the real line are Cauchy-continuous. On the subspace of rational numbers)

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 netss (or equivalently filters is replaced with an arbitrary Cauchy net is Cauchy-continuous if and only if, given any Cauchy filter 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)

See also

• Heine–Cantor theorem

References

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