I am having trouble of understanding what really Scott continuity means. I know the formal definition of what it is, but I don't see why it is the natural way of defining continuity.
Another question is how to interpret it on different kind of sets and order. For example if the set is power set of $A$, ordered by inclusion then a function being Scott continuous would mean $$\forall X \in \mathcal{P}(A) f(\bigcup X)=\bigcup f(X) $$ Could anyone bring more examples of functions which are Scott continuous, on different sets and orders and explain how to interpret it(mainly why it is the natural way of defining Scott continuity)? And also how to use it (the fact of a function being Scott continuous) in different kinds of problems?