Continuity has to do with how things happen over time: if there aren't any bumps or breaks and everything goes on continuously, then there's continuity.
Continuity has to do with consistency. If you have the same Spanish teacher for several semesters, that's great continuity. If you have five different teachers in one year, that's awful continuity. In TV and movies, continuity involves keeping the plot and characters consistent from one scene to the next. It drives fans crazy when a character is allergic to peanuts in one episode, and then eats a peanut butter sandwich in the next one.