Cauchy-continuous function
≈ 1 min read
Names in other languages
No names in other languages are held for this entry yet.
Article bodies are English and are not machine-translated. A name in another language is not a translation of the article.
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.
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. A Cauchy sequence \left(y_1, y_2, \ldots\right) in Y can be identified with a Cauchy-continuous function from \left\{1, 1/2, 1/3, \ldots\right\} to Y, defined by f\left(1/n\right) = y_n. If Y is complete, then this can be extended to \left\{1, 1/2, 1/3, \ldots\right\}\cup\{0\}; f(0) will be the limit of the Cauchy sequence.
Text derived from https://en.wikipedia.org/w/index.php?oldid=1321323212 under CC BY-SA 4.0; modifications have been made.