Continuity mistake: When Katara goes to hug Zuko at the end, she is not wearing gloves on either of her hands, but then a glove appears on her left hand mid-shot, and her right hand also has one once she puts her arms around Zuko. (00:23:04)
Continuity mistake: When Zuko goes inside Sokka's tent to talk to him, Sokka is lying on the floor, but in the next shot, he is a few inches forward from where he was last. (00:07:24)
The Headband - S3-E2
Continuity mistake: When Zuko and Mai are having a picnic, Azula asks Zuko if she can have a word with him. In the next shot, Mai is wearing a black glove on her left hand, which she wasn't wearing in the previous shot. Two shots later, the glove is gone. (00:15:03)
The Tales of Ba Sing Se - S2-E15
Continuity mistake: The Tale of Zuko: When Jin and Zuko are at the restaurant, some of the items on their table constantly disappear, reappear, and change position depending on the shot. (00:16:30)
Continuity mistake: In the shot after Mai says "Saving the jerk who dumped me" you can see the rods which the prison guards used to slow down the mechanism of the gondola are nowhere near the wheel. (00:19:55)
Chosen answer: Toph had Katara use it to send a letter to her parents. It never returned.