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 X and Y be metric spaces, and let f be a function from X to Y. Then f is Cauchy-continuous if and only if, given any Cauchy sequence (x1, x2, …) in X, the sequence (f(x1), f(x2), …) is a Cauchy sequence in Y.
Every uniformly continuous function is also Cauchy-continuous, and any Cauchy-continuous function is continuous. Conversely, if X is totally bounded, then every Cauchy-continuous function is uniformly continuous and if the space X is complete, then every continuous function on X is Cauchy-continuous too. More generally, even if X is not complete, as long as Y is complete, then any Cauchy-continuous function from X to Y can be extended to a function defined on the Cauchy completion of X; this extension is necessarily unique.
Since the real line R is complete, the Cauchy-continuous functions on R are the same as the continuous ones. On the subspace Q of rational numbers, however, matters are different. For example, define a two-valued function so that f(x) is 0 when x2 is less than 2 but 1 when x2 is greater than 2. (Note that x2 is never equal to 2 for any rational number x.) This function is continuous on Q but not Cauchy-continuous, since it cannot be extended to R as a continuous function. On the other hand, any uniformly continuous function on Q must be Cauchy-continuous. For a non-uniform example on Q, let f(x) be 2x; this is not uniformly continuous (on all of Q), but it is Cauchy-continuous.
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