Very short proof that limit preserving functions (continuous functions) on complete partial orders are necessarily monotone.
A complete partial order (hereon cpo) is a partial order such that all monotone chains
has a least upper bound
A continuous function between two cpos and is one that preserves the limit/least upper bound on all monotone chains:
All continuous functions are monotone in the sense that .
Proof: For the sake of deriving a contradiction, suppose that there exists some non-monotone continuous function for some pairs of cpos and . Then it must be the case that for some pair of elements , but . Now, consider an -chain
that is -terminal and monotone. Obviously, its least upper bound , but
because otherwise . Hence, we can show that
and hence cannot be continuous; a contradiction! Therefore, all continuous functions are monotone.