Character mistake: When Jet tells Katara and Aang to waterbend the water from underground in the geysers, Katara states that she has never waterbent on water she couldn't see before. However, six episodes prior to this, in 'The Warriors of Kyoshi', she waterbends the water out from Aang's lungs to save his life.
Suggested correction: That was a moment of desperation on her part, and she probably doesn't think of it/think she can do it again without the same desperate impetus, Furthermore, the water in Aang was only a small amount and very close anyway.
Regardless of if she did it in desperation or not, she did use bending on water she didn't see. The mistake isn't suggesting anything about Katara's ability to waterbend unseen water, the mistake is only in regards to the statement Katara made about never using bending on water she can't see. And for Katara to forget the one time she used bending on water she didn't see would qualify as a character mistake as well. Now if she made the statement "I don't think I can do this" or "I've never used bending on underground water", etc, that would not be a character mistake.
The Boy in the Iceberg - S1-E1
Continuity mistake: In this episode, Katara remarks that "This ship has haunted my tribe since Gran Gran was a little girl. " However, in the episode "The Puppetmaster" it is revealed the ship wasn't beached there until Gran Gran was in her late teens or early twenties.
Suggested correction: You sure that was the same ship, and what's 5-10 years give or take compared to Gran-Gran's life anyway?
