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continuous function
In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output. Otherwise, a function is said to be a ''discontinuous'' function. A continuous function with a continuous inverse function is called a homeomorphism.
Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.
As an example, consider the function ''h''(''t''), which describes the height of a growing flower at time ''t''. This function is continuous. By contrast, if ''M''(''t'') denotes the amount of money in a bank account at time ''t'', then the function jumps whenever money is deposited or withdrawn, so the function ''M''(''t'') is discontinuous.
==History==
A form of this epsilon-delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of as follows: an infinitely small increment of the independent variable ''x'' always produces an infinitely small change of the dependent variable ''y'' (see e.g., ''Cours d'Analyse'', p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasn't published until the 1930s. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.
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