Continuity mistake: When Joker is in the police car, the left eye paint flows across the corner of his mouth. Then the car crashes, people pull him out of the car, and the paint just reaches his nose. Then he stands up and the previous style is back. (01:46:50 - 01:50:10)
Continuity mistake: Between the shot when Joker widens his smile painting it with blood and the following wider angle, the items people in clown mob are waving around change suddenly. (01:52:10)
Continuity mistake: Murray's blue jacket, when he comes out from behind the curtain his jacket is open, than he makes a twirl and the first button of the jacket is buttoned up.
Continuity mistake: Arthur's cuffed in the police car, but he's uncuffed once laid out on the hood.
Continuity mistake: At the end when Arthur/Joker dances on the police car's hood in front of the crowd, in the shot behind him he keeps his arms down, but when it changes into a frontal shot on him he keeps them lifted up.
Continuity mistake: When Joker confesses that he was the one who killed the three guys in the subway at the Murray Franklin Show and also says "It's not a joke", he keeps his gaze turned towards Murray in one shot and towards the audience in the next.
Answer: Yeah it's completely up to the viewer to believe he killed her or not. I don't think he did, he liked her, just like Gary. I think he visited to see if it was all in his head, with that confirmed he just left.
lionhead