Let [ ] : Ord -> Relation Nat be a function that realizes an ordinal into a well-ordering of the naturals with that order type.
Given a relation R on the naturals, we define the restricted relation R(<n) to be R but just acting on the first n naturals.
Defn: [ ] is continuous iff for every series of ordinals a_i whose limit is l, for every natural n, there exists an i
such that for all j > i, [a_i](<n) = [l](<n).
Intuitively, if you are only looking at a finite prefix of the naturals, you can get to a limit ordinal by going finitely high up its limit sequence.