continuous function

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noun Mathematics.

(loosely) a mathematical function such that a small change in the independent variable, or point of the domain, produces only a small change in the value of the function.

(at a point in its domain) a function that has a limit equal to the value of the function at the point; a function that has the property that for any small number, a second number can be found such that when the distance between any other point in the domain and the given point is less than the second number, the difference in the functional values at the two points is less than the first number in absolute value.

(at a point in a topological space) a function having the property that for any open set containing the image of the point, an open set about the given point can be found such that the image of the set is contained in the first open set.

(on a set in the domain of the function or in a topological space) a function that is continuous at every point of the set.

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Based on the Random House Dictionary, © Random House, Inc. 2012.
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Computing Dictionary

continuous function definition

A function f : D -> E, where D and E are cpos, is continuous if it is monotonic and
f (lub Z) = lub f z | z in Z
for all directed sets Z in D. In other words, the image of the lub is the lub of any directed image.
All additive functions (functions which preserve all lubs) are continuous. A continuous function has a least fixed point if its domain has a least element, bottom (i.e. it is a cpo or a "pointed cpo" depending on your definition of a cpo). The least fixed point is
fix f = lub f^n bottom | n = 0..infinity
(1994-11-30)

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