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continuous function : FOLDOC | continuous function 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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