Editing Continuous function (section)

{{Short description|Mathematical function with no sudden changes}}
{{Calculus}}

In [[mathematics]], a '''continuous function''' is a [[function (mathematics)|function]] such that a small variation of the [[argument of a function|argument]] induces a small variation of the [[Value (mathematics)|value]] of the function. This implies there are no abrupt changes in value, known as ''[[Classification of discontinuities|discontinuities]]''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A '''discontinuous function''' is a function that is {{em|not continuous}}. Until the 19th century, mathematicians largely relied on [[Intuition|intuitive]] notions of continuity and considered only continuous functions. The [[(ε, δ)-definition of limit|epsilon–delta definition of a limit]] was introduced to formalize the definition of continuity.

Continuity is one of the core concepts of [[calculus]] and [[mathematical analysis]], where arguments and values of functions are [[real number|real]] and [[complex number|complex]] numbers. The concept has been generalized to functions [[#Continuous functions between metric spaces|between metric spaces]] and [[#Continuous functions between topological spaces|between topological spaces]]. The latter are the most general continuous functions, and their definition is the basis of [[topology]].

A stronger form of continuity is [[uniform continuity]]. In [[order theory]], especially in [[domain theory]], a related concept of continuity is [[Scott continuity]].

As an example, the function {{math|''H''(''t'')}} denoting the height of a growing flower at time {{mvar|t}} would be considered continuous. In contrast, the function {{math|''M''(''t'')}} denoting the amount of money in a bank account at time {{mvar|t}} would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.