Cauchy-continuous Function - Examples and Non-examples

Examples and Non-examples

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 can't be extended to ℝ. 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.

A Cauchy sequence (y₁, y₂, …) in Y can be identified with a Cauchy-continuous function from {1, 1/2, 1/3, …} to Y, defined by f(1/n) = y_n. If Y is complete, then this can be extended to {1, 1/2, 1/3, …, 0}; f(0) will be the limit of the Cauchy sequence.