a real world point of view
A continuity supervisor or script supervisor — often credited under the shortened title, continuity — is a person tasked with ensuring that scenes filmed out of narrative order make sense when edited into the proper sequence. They thus have to watch carefully as principal photography continues, and take detailed notes as to the daily progress of the film unit. They ensure that actors render the script as precisely as is needed to convey the script writer's intended meaning. They also keep an eye on visual details such as costume, make-up and props to maintain the narrative flow of the story, even though it may be filmed out of order.
For instance, if the script calls for several versions of the same prop to be used at different stages in the script, the continuity supervisor is ultimately responsible for making sure the actor is holding the correct version of the prop as she says particular lines. The continuity supervisor is also responsible to making sure the actor remembers the emotional "level" of their character at a given moment of a script so that the tone of a building emotion can be preserved across a scene when edited together. In rarer instances, the script editor, who must be familiar with the script at a detailed level, may notice inconsistencies in the script itself, and may recommend on-set changes to the script editor, writer or director.
The post has taken on increasing importance throughout the history of the production of Doctor Who. In the 1960s, the show was filmed more like a play than a film. Editing was infrequently allowed. Thus, the script supervisors of the day would've only intervened when an incident occurred that would've rendered the captured footage entirely unusable. The task was often performed not by a specialist, but by an assistant director or floor manager. As the show moved to a more disjointed style of production, the role of the continuity supervisor greatly increased. They were particularly useful in matching up scenes partially recorded on location and those completed in studio. In the current BBC Wales version, which is regularly made of around a thousand individual fragments, they are invaluable in keeping order on the set.