24 **M. Slota and J. Leite**
Proof Suppose S = { s1, s2, . . . , sn }**. Then we can construct a sequence** {di}
n i=2 **of elements**
of D **such that**
s1 ≤ d2 and s2 ≤ d2 ;
si ≤ di and di−1 ≤ di for each i ∈ { 3, 4**, . . . , n** } .

By induction on i it follows that di ≤ dn for every i ∈ { 2, 3, . . . , n } **and by applying** transitivity we obtain si ≤ dn for each i ∈ { 1, 2, . . . , n }. Hence dn **is an upper bound**
of S in D.

As an important consequence, we obtain that every finite directed **set contains**
its own supremum.

Corollary A.7 Any finite directed set contains its supremum. Proof Let (D, ≤) be a finite directed set. Then by Prop. A.6 **it contains its own upper** bound d. Consider some other upper bound u of D. Then since d ∈ D**, we have** d ≤ u and so d is the least upper bound of D**, i.e. the supremum of** D.

We can now introduce two properties of functions on partial orders. the weaker property of monotonicity basically states that the function preserves the partial order:
Definition A.8 (**Monotonic Function**) Let P, Q be two partially ordered sets and f : P → Q. We say f is monotonic **if for**
every a, b ∈ P such that a ≤ b **we have** f(a) ≤ f(b).

The property of continuity is stricter and requires that for all directed sets with a supremum in the domain, the image of that supremum is the same as the supremum of images of elements of the directed set.

Definition A.9 (**Continuous Function**)
Let P, Q be two partially ordered sets and f : P → Q. We say f is **continuous** if for every directed subset D of P with supremum in P **it holds that**
sup(f(D)) = f**(sup(**D))
where f(A) = { f(a) | a ∈ A } **for any set** A ⊆ P.

The next proposition formally proves that continuity is a stronger property thatn monotonicity.

Proposition A.10 Every continuous function is monotonic.