Continuity mistake: Before Alice enters the building, you can see part of her newspaper sticking out of her bag - the area that is visible just contains text. After she enters the building, there is another shot of her from behind, and now the paper is showing part of a picture and the barcode.
Continuity mistake: In the scene after Heather Graham has gone to Joseph Fiennes' house and they start to have sex, Finnes carries Heather into a room and places her on the floor, in this shot Heather's bra is tucked below her breasts, in the next her breasts are covered, in the next they are exposed again.
Continuity mistake: When Alice and Adam get in the cab outside the bookshop, the number on the back of the taxi is 90167, there are no other markings on the back and the Y on the number plate has a long stalk. When they're pulling away from some traffic lights later, the number has changed to '14290', there are the words "fairaway 95" on the back and the 'Y' has a small stalk. Note that, since the number plate (K651EYW) stays the same, for at least one of the shots they were driving a car around London with the wrong plates on.