When Faye (Faye Wong) delivers a letter to Policeman 663 (Tony Leung Chiu-Wai), said letter is being held in a way that makes the front side point down. However, in the following cut, it is held straight.
In the part where Faye leaves the apartment and the camera shows her going out, a portion of the camera is seen in the mirror for a brief moment.
The woman in the blonde wig had no apparent opportunity to get the beeper number except from Cop 223 himself the night before, and he wouldn't have started to abandon his beeper as useless if he had just given someone new the number.