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).