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 fixedpoint 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)