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.
is Cauchy-continuous if and only if, given any Cauchy sequence
Every uniformly continuous function is also Cauchy-continuous.
is totally bounded, then every Cauchy-continuous function is uniformly continuous.
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.
is complete, then every continuous function is Cauchy-continuous.
is complete, then any Cauchy-continuous function from
can be extended to a continuous (and hence Cauchy-continuous) function defined on the Cauchy completion of
this extension is necessarily unique.
is compact, then continuous maps, Cauchy-continuous maps, and uniformly continuous maps on
is complete, continuous functions on
of rational numbers, however, matters are different.
For example, define a two-valued function so that
but not Cauchy-continuous, since it cannot be extended continuously to
On the other hand, any uniformly continuous function on
can be identified with a Cauchy-continuous function from
will be the limit of the Cauchy sequence.
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.
is Cauchy-continuous if and only if, given any Cauchy filter
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.
may be made into a Cauchy space.
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