by Craig Kurtz
Continuity is the principle of making sure that all details in a film are consistent from shot to shot & from scene to scene.    If a scene upholds the standards of continuity, each shot feels as though it seamlessly flows from the previous shot reinforcing the sense of realism in the shot.     The SCRIPT SUPERVISOR is the person responsible for making sure this is established in the film.    However, in many films including well-known productions, a lack of continuity can be found.   Sometimes this is due to budget and deadline constraints It may have been missed initially upon review and when the editing is done, the Producers decide to leave it in the film especially since this will involve additional cost… and it may not even be noticed by film-goers.   I enjoy watching films many times and begin to look at the details of the film after my first couple of viewings. Check out the examples of continuity problems from several films. 
Enjoy the interesting detail of a film and the many, many facets of making a film. 
The Wizard of Oz - Dorothy meets the Scarecrow - sync w/Pink Floyd's 'Brain Damage' and 'Eclipse' - YouTube
Wizard of Oz – When Dorothy meets the Scarecrow, her hair changes length several times as she interacts with Scarecrow on Yellow-brick Road.

Pulp Fiction (1994) - IMDb

Pulp Fiction – In the scene with John Travolta and Samuel L Jackson, the bullet holes are already in the wall in a shot before the shots are fired.   Look closely at the wall.
What Did Jenny Die From In Forrest Gump? Her Cause of Death
Forrest Gump –  Forrest goes to Jenny’s apartment, the shot clearly shows the iron on the ironing board is sitting up, in the next immediate shot the Iron is laying flat.   Again the scenes can be shot at different times or maybe something happened as they were filming additional takes of the scene.
Harrison Ford Doesn't Share Indiana Jones' Greatest Fear
Indiana Jones (1st movie)  Indiana falls into the room of snakes.  He ends up face to face with a Cobra which flares its Hood.  But in order for Harrison Ford to be safe in the scene, a piece of glass separates the live Cobra from Harrison Ford.   But look closely and you will see the reflection of the Cobra in the glass.
The 'Star Wars: Empire Strikes Back' Mistake Hiding In Plain Sight For 39 years | Flipboard
Star Wars  When Luke is taken to the Wampa Cave and is hung upside down, his Light Saber falls straight up into the snow.  If you look closely, you will see the words New York written on the bottom of the Saber.  New York is where the Saber was made and to my knowledge does not exist in the Star Wars story or maybe we are just another part of the galaxy.
Again remember that when filming they are shooting the same scenes on different times of day, different days or different seasons.  So the Script Supervisor must make sure these things look and feel the same.
So while some may not care about the details in a film, it is all there for you to absorb and has an impact on your viewing.   And of course, it makes all the difference to great filmmakers who inspire everyone who watches the film