Continuity mistake: When Pete awakes on the shore, the dog licking his face steps over his head, leaving footprints in the sand. In the following shots, however, the footprints have disappeared. (00:18:40)
Continuity mistake: After Jim Carrey crashes the car on the bridge and, we see in the long shots that the lamp on the half-broken post above him is swinging from side to side. However each time we cut to a closeup of Carrey's face, it is lit very evenly - although the only source of light nearby is swinging wildly.
Continuity mistake: When Jim Carrey's car goes into the water, it lands upside down. Then he is underwater and gets his coat off. The car is completely submerged at this point. When he comes up for air, the car is only halfway submerged with one of the ends sticking up perpendicular to the surface of the water.
Answer: The father showed a picture of Luke. Peter and Luke were identical twins.
I'm hoping that this answer is implying that Peter and Luke are spitting images, and not actual identical twins, as there is no relation between the two men. Adele had her theories that Peter was not Luke, and Emmett stated that he knew at the reunion party. The town is fooled for a couple of reasons, mainly because he looks just like him. But also because they wanted to believe it was him. The town had suffered a lot of heartbreak after the war, and this was some good news, while it lasted.
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