Continuity mistake: In the scene where Malcolm Crowe is showing Cole the "vanishing penny" trick, watch Malcolm's hair as the scenes cut back and forth between the child and Malcolm. Malcolm's hair (front) goes from perfectly combed to a duck tail. (00:38:30)
Continuity mistake: The tape is labelled "Vincent Grey" in the first shot, but the label disappears in the next shot. (01:11:32)
Continuity mistake: During the "I See Dead People" scene, Cole is facing Malcolm and a tear runs down his face. When Cole turns away, his face is dry with no trace of a tear.
Continuity mistake: In the scene where Cole and his mother are sitting in traffic, the camera closes in on Cole as he states: "She's standing next to me." The next shot of Cole shows the accident victim standing behind him. However, the background has totally changed, and the lighting is totally different. This second shot was probably done on a set instead of the actual street.
Continuity mistake: In the scene where Lynne Sear is collecting laundry, she stops to look at the family photos on the hallway wall and she is wearing orange lipstick. When she leaves the hallway and walks into Cole's room to pick up his dirty clothes, she is wearing no lipstick.
Continuity mistake: Before Dr. Crowe first speaks with Cole, he is sitting reviewing notes and when Cole comes out of the apartment, Crowe flips to the notes on Cole. The camera is in tight, and moves to close ups of words on the page. Later when translating the Latin Cole was saying, we see a large part of the page, and it is obviously the same page from earlier, including the words that were zoomed in on. Problem is it starts by describing how they first met in a church, which when Crowe was first reviewing the notes had not yet happened. It also describes Cole saying things we don't hear him say in the church. It is obvious from the greeting that the meeting in the church we see is the first time they meet, so how did Crowe have a written description of an event that had not yet happened? There is no indication that ghosts forget events they do after they died, so it wouldn't make sense that this isn't the first meeting, and Crowe just doesn't remember.
Continuity mistake: When Cole enters Kyra's room, he enters through a door on the left side of her bed. When her dad watches the tape, it shows her mom enter the right side.
Answer: Because as Cole tells us in the middle of the movie, "They only see what they want to see." The movie shows us Malcolm as he perceives himself. He can't actually put on an overcoat, or change clothes. He sees what he expects to see. He goes outside in the cold to visit Cole, so he sees himself wearing an overcoat. He doesn't know he is dead - he doesn't see his wound. To preserve this misdirection throughout the movie, we are shown Malcolm as he sees himself.
Actually, we do see Malcolm and Cole sees him. The entry wound in his abdomen is tiny, and the large, obvious exit wound is on Malcolm's back, which we never see.
C Classic