Continuity: In the scene where you see the children sliding down Appa's tail, in the close up shot it's leaning on the sticks. From far away it changes to lying on the ground, then close up again, it's back on the sticks.
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Updates: 
Series 1
The Boy in the Iceberg (season 1, episode 1)
The Fortuneteller (season 1, episode 14)
Plot hole: Because Katara lost her mothers necklace, Aang made one for her and she puts it on. But when they reach The Fortunetellers village, Katara is not wearing the necklace, and no one notices, not even Aang.
Avatar Day (season 2, episode 5)
Continuity: When the village leader switches the wheel of punishment to Community Service, it is on the space next to Boiled in Oil. However, all of the other shots of the wheel have another section in between these two.
The Desert (season 2, episode 11)
Factual error: When Aang enters the avatar state, his eyes and forehead arrow glow like normal, but the arrows on his hands do not.
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