Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous. An intuitive though imprecise (and inexact) idea of continuity is given by the common statement that a continuous function is a function whose graph can be drawn without lifting the chalk from the blackboard.
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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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