In mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

s (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

Let

X

and

Y

s, and let

f : X \to Y

be a function from

X

Y.

Then

f

is Cauchy-continuous if and only if, given any

Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...

\left(x_1, x_2, \ldots\right)

X,

the sequence

\left(f\left(x_1\right), f\left(x_2\right), \ldots\right)

is a Cauchy sequence in

Y.

Properties

Every uniformly continuous function is also Cauchy-continuous. Conversely, if the domain

X

totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “si ...

, then every Cauchy-continuous function is uniformly continuous. More generally, even if

X

is not totally bounded, a function on

X

is Cauchy-continuous if and only if it is uniformly continuous on every totally bounded subset of

X.

Every Cauchy-continuous function is continuous. Conversely, if the domain

X