During the filming of the Ludovico scene, Malcolm McDowell actually had one of his corneas scratched. He states that he didn't realize it until he was leaving the set that night and the anesthetic wore off. He also states that it felt like someone hammered a nail into his eye. See more...
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Continuity: The main character's name is several times in the film mentioned as 'Alexander DeLarge', but in the newspaper articles detailing his release from prison, they call him 'Alex Burgess'. Burgess is the surname of the author of the book the film is based on, so they might have originally intended to name him so as an homage.
Continuity: When Alex, the writer, and the other two political people are sitting at the table near the end of the movie, not only does the amount of sauce change on the spaghetti, but the amount of spaghetti changes from shot to shot even after Alex has stopped munching on it. This isn't part of the director's ploy.
Continuity: When Alex and his 'droogs' are driving up to the 'Home Sweet Home' house, in the background you can see a figure running to a tree. It's very dark and hard to see, so it could admittedly be a horse or something; but it is there. [It is a horse, so not a mistake as such. Still worth looking for].