Without break or step. Continuity is what makes a later moment belong to the same thing as an earlier one.
Latin continuus (unbroken, connected), from continere (to hold together), from com- (together) + tenere (to hold); without break or step, each moment holding directly onto the next.
It is the adjective form of the same holding that continuity names as a noun, not a state a thing has but a way a thing simply is, moment against moment with nothing between them. Where continuity is the thread, continuous is what a thing looks like made entirely of thread.