Cauchy-continuous function
Encyclopedia
In mathematics
, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function
between metric space
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.
s, 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.
. Conversely, if X is a complete space
, 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.
ℝ is complete, the Cauchy-continuous functions on ℝ are the same as the continuous ones. On the subspace ℚ of rational number
s, 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) = yn. 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.
. 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 on Y. This definition agrees with the above on metric spaces, but it also works for uniform space
s and, most generally, for Cauchy space
s.
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/∞.)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
between metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
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 be metric spaceMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s, and let f be a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
from X to Y. Then f is Cauchy-continuous if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
, given any Cauchy sequence
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...
(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, and any Cauchy-continuous function is continuousContinuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
. Conversely, if X is a complete space
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
, 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.
Examples and non-examples
Since the real lineReal line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
ℝ is complete, the Cauchy-continuous functions on ℝ are the same as the continuous ones. On the subspace ℚ of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, 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) = yn. 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.
Generalisations
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 netCauchy net
In mathematics, a Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.A net is a Cauchy net if for every entourage V there exists γ such that for all α, β ≥ γ, is a member of V. More generally, in a Cauchy space, a net is Cauchy if the filter generated by the...
. 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 on Y. This definition agrees with the above on metric spaces, but it also works for uniform space
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
s and, most generally, for Cauchy space
Cauchy space
In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to...
s.
Any directed set
Directed set
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
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/∞.)