continuous function

Function such that the preimage of an open set is open

Entitytype of mathematical functionQ170058

Also known as continuous map · continuous application · continuous transformation

Instance of: type of mathematical function
Subclass of: piecewise continuous function

Overview

From Wikipedia, the free encyclopedia · Read full article · Text CC BY-SA 4.0

In mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting its argument to sufficiently small changes. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

Overview

All data

38 statements · 21% with external sources · numbers in brackets are citations

Has characteristic: continuity
Defining formula: f:(X,T_1)\to (Y,T_2)\quad\and\quad\forall\,U\in T_2:\quad f^{-1}(U)\in T_1
Different from: real-valued continuous function of a real variable
continuity
Said to be the same as: continuous function
Opposite of: discontinuous function
discrete function

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27 identifiers linking continuous function across the web's reference databases

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Source last edited August 13, 2026. AIDB refreshes this page daily.

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AIDB. (2026). continuous function (Q170058). Retrieved from https://aidb.si/e/continuous-function-Q170058