Is continuity a team effort, with the costume designer, for example, responsible for costume continuity, the standby props for action props and so on? Two dangers are inherent in this idea.
Firstly, it is confusing the planning of a production with the shooting. For example, during the planning the costume designer has noted that Scenes 1–4 had direct continuity, and decides that the old tramp appearing in each scene should wear a coat, tied at the waist with string, and a battered hat. When you come to shoot Scene 2, the first in shooting order, the director decides that the hat and coat should be left behind on a park bench. When Scenes 3 and 4 are therefore shot, later in the shooting schedule, the tramp must be without his coat and hat.
But supposing the costume designer was not able to attend the shooting of Scene 2? There might have been a costume fitting to supervise or another urgent appointment. It is also possible that there was not room for all the crew to be present at the shooting due to lack of space.
This is not to denigrate the planning done by others in any way. A great deal of assistance is given to continuity by the different members of the unit. But your concern is not with what was planned but what was actually shot, scene by scene and day by day.
The second danger is that everyone sees things differently from the viewpoint of their own particular interest in the production. Costume will notice costume details to the exclusion of all else, props will notice props, the camera operator will note the composition and framing of the shot, and so on. You are the only person whose responsibility is not to any specific aspect of the shot but to the overall picture, and your concern is how that shot in its entirety relates to others in story order. You are objective, standing in, in a sense, for the editor and the viewer. You look at all the ingredients that make up the composition of a shot and decide which should have priority over others.
‘Costume will notice costume details to the exclusion of all else’.