Analyze the Changes and Continuities in Long - amazonia.fiocruz.br

In mathematicsa continuous function is a function that does not have any abrupt changes in valueknown as discontinuities.

More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous.

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Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon—delta definition were made to formalize it. Continuity of functions is one of the core concepts of topologywhich 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. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces.

In order theoryespecially in domain theoryone 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, the function H t denoting the height of click growing flower at time t would be considered continuous.

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In contrast, the function M t denoting the amount of money in a bank account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn. A form of the epsilon—delta definition of continuity was first given by Bernard Bolzano in Cours d'Analysep. Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today see microcontinuity.]