Factual error: When introducing himself to the Toclafane, President Winters refers to himself as "President-elect of the United States of America." President-elect is the title given to the victor of an American presidential election who has not yet been sworn in, between early November and January 20. This was a mistake on Russell T Davies' part, as he was under the impression it was a longer, formal title for the American president. (00:37:40)
Continuity mistake: When the Master stands up on the Valiant to announce that he is the Toclafane's "Master", President Winters is visible in the background, only the Toclafane aren't there. In the next shot, they return, buzzing around Winters' head.
Continuity mistake: When Martha starts to phone her family, after the Doctor warns her not to tell them anything, as she snaps in response, "I'll do what I like!", in the first shot she's holding her phone away from her body, but in the second shot it's jumped to her ear. (00:14:25)
Continuity mistake: When the Doctor fixes Jack's vortex manipulator so they can escape the silo, Martha and Jack are holding the lab door mostly closed against a horde of Futurekind. In the shot where the trio time-travels away, the Futurekind, some of whom had stuck their hands through the gap in the door, are missing. (00:00:40)
Continuity mistake: When the Doctor talks about how the Master was always sort of hypnotic, in a wide shot, Martha is sitting on the end of her couch, with her hands loosely folded, with her left hand on top. In the shot of her where she says she was going to vote for Harold Saxon, her right hand is on top. In the next wide shot, her hands have reverted to their original position. (00:12:05)
Continuity mistake: On the Valiant's bridge, when the Doctor is telling Martha and Jack about his plan to slip one of the perception filter keys around the Master's neck while holding his key, the position of the key changes in his hand, from being held by the wide end to being held by the narrow end. (00:37:15)