Continuity mistake: In the scene toward the end where the Doctor is shot by the Dalek, when the camera is shooting from Rose's point of view, the Doctor is gasping and struggling for breath. However when the camera shows the Doctor from behind, he's lying completely still in Rose's arms. The camera cuts back to Rose's POV and he's struggling again, with no time in between for the change in his movements.
Factual error: The Doctor (and the staff on the station) refer to the idea of a planet being in orbit around a black hole as 'impossible'. It is not. Gravity (and physics in general) works perfectly well outside of the event horizon. For all practical purposes - regarding orbiting around it - the black hole might as well be any other object, as long as it has an equal mass.
Plot hole: In this episode, the Doctor is said to have died during the events of "The Runaway Bride." As a result, he was not able to prevent the spaceship Titanic from crashing in London. It is later shown that southeast England was obliterated by the explosion. This falls far short of what the Doctor mentioned in "Voyage of the Damned", wherein he frantically repeated that if the Titanic crashed, the drive's explosion would wipe out all 6 billion people on the Earth (that's the entire human race at the time of the show's airing).
Other mistake: Martha falls into the bog and Hath Peck rushes down to help her. She calls out "Help me, Peck!" but the subtitles read "Help me peg!" This is just one of many errors in this series' subtitles, which seem to have been added by listening to the dialogue without reference to the script.
Continuity mistake: When Ursula is filming Elton outside his old house, the digital camera produces a distorted picture and has four white lines in the corners. But, after Elton's flashback, Ursula is still filming but the picture is straight and the lines are gone.
Revealing mistake: Whenever you glimpse the inside of the TARDIS from outside, a frontal view reveals a picture of what's inside which has been placed at the back of the box. Sadly, it isn't on the sides. This means that, from the right angle, you can easily tell that it is just a box.
Factual error: Near the end of the episode, the full Moon is shown in the daytime sky, well above the horizon. This is impossible, as any celestial body lit by the Sun has its full phase only when it is directly opposite the Sun; thus, a full Moon rises at sunset and sits high in the sky only during the night. For the Moon to be in the sky during the day as shown, it would have to be visibly of a phase other than full.
Other mistake: When the Doctor finds the crystal and blows on it, it glows brightly. While it lights up his face as it should, the backs of his hands light up, apparently from the crystal. This is not possible, as the light comes from in front when the crystal is inside the Doctor's hands.
Factual error: The rank insignia of the Soviet officers is completely inaccurate. The captain wears the epaulets of a captain first rank (captain) and the sleeve insignia of a captain second rank (commander). Lieutenant Stephashin wears the epaulets of a senior lieutenant and the sleeve insignia of a captain third rank (lieutenant-commander). The third officer wears the epaulets of a captain second rank and the sleeve insignia of a captain third rank.