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.

Definition

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.

Properties

Every uniformly continuous function is also Cauchy-continuous. Conversely, if the domain X is totally bounded, 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 is complete, then every continuous function is Cauchy-continuous. 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 continuous (and hence Cauchy-continuous) function defined on the Cauchy completion of X; this extension is necessarily unique.

Combining these facts, if X is compact, then continuous maps, Cauchy-continuous maps, and uniformly continuous maps on X are all the same.

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 cannot be extended continuously 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 2^x; this is not uniformly continuous (on all of Q), but it is Cauchy-continuous. (This example works equally well on R.)

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.

Generalizations

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 (x₁, x₂, …) is replaced with an arbitrary Cauchy net. Equivalently, a function f is Cauchy-continuous if and only if, given any Cauchy filter F on X, then f(F) is a Cauchy filter base on Y. This definition agrees with the above on metric spaces, but it also works for uniform spaces and, most generally, for Cauchy spaces.

Any directed set A may be made into a Cauchy space. Then given any space Y, the Cauchy nets in Y indexed by A are the same as the Cauchy-continuous functions from A to Y. If Y is complete, then the extension of the function to A ∪ {∞} will give the value of the limit of the net. (This generalises the example of sequences above, where 0 is to be interpreted as 1/∞.)

gollark: ```pythonprint("Hacked with python 3")```

gollark: ```print "Hacked with Python 2 or Lua"```

gollark: (produced by the common Unix tool `haxxdump`)

gollark: 011d3b0 ecda fe42 f33d d112 2b8c 7e1d 24d2 11e5011d3c0 2475 ae6a bb0f 0c59 592b 3e75 6074 5f61011d3d0 ff42 a907 c773 c81f 3095 97ba 7fe2 5270011d3e0 c021 d886 1dfc 01eb f22a 0174 38cb ab3e011d3f0 2476 6efa 2bb0 6dde cd92 0222 5467 7221011d400 bb13 2647 77f7 8c51 6206 e40d 3c85 117c011d410 86bb 928f 2234 bb31 298e dd89 7209 6a00011d420 49b1 182b 52fc 6659 f720 c14c 7064 213c011d430 be13 5b7f 36db 9228 232a be39 1c9e 4065011d440 3e92 3fa8 a538 8a60 c599 7c88 9f72 9748011d450 8a5d fc83 b21b e48d 666a 8670 3d61 0225

gollark: I have made many a useless side project.

References

Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York.

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