## 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}