A function f may have many fixed points (x such that f x = x). For example, any value is a fixed point of the identity function, (\ x . x). If f is recursive , we can represent it as
f = fix F
where F is some higher-order function and
fix F = F (fix F).
The standard denotational semantics of f is then given by the least fixed point of F. This is the least upper bound of the infinite sequence (the ascending Kleene chain ) obtained by repeatedly applying F to the totally undefined value, bottom. I.e.
fix F = LUB bottom, F bottom, F (F bottom), ... .
The least fixed point is guaranteed to exist for a continuous function over a cpo .