
Continuity mistake: Toph mentions the worst thing about being disguised is they don't get the hero worship anymore and she misses the love, to which Sokka grumbles, "Boo-hoo, poor heroes." Then, as Sokka says, "It's just, all you guys can do this awesome bending stuff," the triangular emblem on Aang's red "headband" has vanished, and in the next shot, it reappears. (00:03:35)

Continuity mistake: When Piandao tells Sokka that creativity, versatility, and intelligence define a great swordsman, he holds out Sokka's sheathed sword with its hilt at Piandao's right hand. However, in the next shot facing Sokka, the sword in its scabbard now lies on Piandao's hands the opposite way, with its hilt at Piandao's left hand. The position of Piandao's hands and thumbs changes as well, between shots. (00:17:25)

Other mistake: When Aang is reading the letter from The Guru, as Sokka says, "What's a guru? Some kind of poisonous blowfish?" his right eye loses its white colour three times during the shot. (00:18:19)

Continuity mistake: After Meng gives Aang the tray of food and Sokka takes it from him, Aunt Wu shows up, and it shows a shot facing her front. In this shot, Katara's back braid is missing; then it reappears a moment later. (00:07:05)

Sozin's Comet (1): The Phoenix King - S3-E18
Continuity mistake: When Aang is waking up on the giant lion turtle, he is not wearing shoes in the first shot, but in the next shot, he is. (00:22:35)

Bato of the Water Tribe - S1-E15
Other mistake: When Katara is behind Sokka and he is speaking she leans back and as she does her eyes stay in the same place while her head moves, then it fixes itself. (00:16:20)

Continuity mistake: When Katara is trying to heal the wounded earthbender, she gives Sokka the idea to take down The Drill from the inside. Sokka then stops her from saying anything while she is sitting down, and she is wearing her necklace. When she stands up in that same shot, the necklace is gone, like they forgot to animate it. (00:07:35)

Continuity mistake: When Aang emerges from the water on the giant sprout in The Avatar State, the arrow on his head is not glowing.
Chosen answer: When Sozin says "So I wiped out the Air Temples," we are indeed seeing the three mountains of the Eastern Air Temple burning, with the two bridges being gone in this shot (S3-E6). This does conflict a bit with the image we see of the Eastern Air Temple, with the two bridges intact, when Aang and Appa fly to that Air Temple seeking Guru Pathik (S2-E19). This may possibly be considered a mistake, but the fact is that when we are seeing the burning of the Eastern Air Temple it's from the point of view of Zuko, who is merely reading from Sozin's autobiographical account, and envisioning everything he's reading, which in his mind includes the bridges having been destroyed.
Super Grover ★