Continuity mistake: After the boy's car gets crushed, Bart sells Milhouse's glasses for $600. Then Milhouse knocks over the Coke bottles and the clerk gets the money back and keeps the glasses. The next scene, where they try to get on a train that goes to Springfield, you can see Milhouse in his glasses and throughout the rest of the episode.
Factual error: During Dwight's attempted bank robbery, he gets hit with a couple of dye packs, then appears a few shots later totally clean. The dye packs banks use are notoriously difficult to wash off once they're triggered; even if Dwight did manage to clean himself up, there would still be remnants of dye on him.
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Continuity mistake: In the new opening sequence (season 20), you see Kearney and Jimbo sawing off the Jebediah Springfield statue's head, which starts falling down. In this wide plan, overhead and closing up, the whole town square is in full sight with no one visible besides those characters. The shot cuts to a close-up of the statue's head falling on Ralph Wiggum, now right next to the statue's pedestal where he was calmly eating an ice cream cone.