Continuity mistake: As Nicolas Cage is smoking a cigarette in his car when his daughter comes into view for the 1st time. The camera angle flips between him, him with her in the background, to his key, back to him, and back to him with her in the background. Each time the angle changes so does the length of his cigarette. The time span is only a matter of 10-15 secs as she walks up and then starts to walk away, but his cigarette ends up being longer than when she walked up.
Other mistake: As Roy meets his daughter, he gets out of his car and we see the car door left open as he runs towards her. Near the end of their conversation, the camera focuses on Roy, and through the space between his body and right arm, you can see the side view mirror indicating the door is now shut without anyone touching it.
Continuity mistake: Cage's partner removes his clip-on tie when he returns to the office after posing as FBI. As soon as he pulls it off and you see his top button still buttoned, the shot changes and you see his arm movement from pulling it off. But his top button is now unbuttoned.