When Mary and Norman go into Mother's room, before Mary turns on the light look at the shadow on the wall on the right. It is a silhouette of Alfred Hitchcock, the director of the original "Psycho" (1960). See more...
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Continuity: When the two teenagers are down in Norman's fruit cellar, and a "lady" with a knife is chasing the boy, he leaves finger markings on the fogged up window. But by the next shot they have changed. You can tell because when the boy made the markings, his thumb and his index finger markings intersected, but when the camera angle changes the finger prints don't intersect and they are noticeably straightened also.
Audio problem: When Norman is talking to his "Mother" on the phone, the camera (in Mary's point-of-view) comes in the room and Norman says, "How could you think such a thing. I would never do such a thing," but you can see Norman's reflection in the big mirror hanging in the lounge room and you can see his mouth isn't moving.
Continuity: When Norman arrives home, he is startled by a note that is found beside the phone and he accidentally knocks his suitcase down the stairs and most of his clothes land on the top of the stairs. But as the camera angle switches to the bottom of the stairs, the clothes are further down the staircase. They couldn't have moved as they were stationary at the end of the other shot.
Revealing: When Toomey is at the motel packing his clothes after Norman fired him, he is startled by "someone" and he turns around and gets slashed with a knife in the face. Before Toomey gets slashed though, you can see a prosthetic scar on his face and a wire in the top right hand corner to trigger the prosthetic scar to open up and look like a knife wound.