Visible crew/equipment: After the acting Prime Minister gives orders to the sergeant and the lift doors close, the soldier turns around. As he issues orders, the boom mike is reflected on his goggles. (00:05:25)
Character mistake: The Doctor comments that the destruction of Earth will cost 5 billion lives. At the time of the episode (2006, one year on from when Rose met the Doctor), the population of Earth was around 6.5 billion. (00:25:10)
Continuity mistake: Partway through the episode, there's a wide shot of London, with an undamaged Big Ben visible. The inciting incident of the previous episode was a spaceship crashing into the clock tower, severely damaging it. (00:25:40)
Revealing mistake: At the end of the episode there is a side shot of the Doctor leaving the TARDIS in which you can see inside, but the control room isn't visible, just the side of the police box, with light from outside shining through the window. (00:38:00)
Continuity mistake: During the opening montage, showing the Slitheen incapacitated by electrocution, when Margaret drops Indra's body and stumbles back, a close-up of the body on the floor shows her legs without the pulsing electricity covering her entire body, which is visible in other shots.
Chosen answer: The Master knows that deep down, he deserves death for the crimes that he's committed throughout his life, and since he regards The Doctor as his arch-foe, he expects it to be at his hands. The fact that The Doctor is still willing to forgive him for all of his crimes hurts him more deeply than death would.
Captain Defenestrator