Cauchy-continuous function

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. 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 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. Since the real line is complete, the Cauchy-continuous functions on are 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.