Revealing mistake: The Doctor and Rose locate the "metal" hatch that leads down into the Nestene Consciousness' lair. Watch carefully after the Doctor removes the "metal" hatch and moves it aside. The entire hatch pops up a second later. This is because it is plastic and it was suctioned to the floor.
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Continuity mistake: When Rose presents her mother with the Bazoolium, she's holding it in her left hand. When the view switches to look at Jackie, it shifts to Rose's right hand and then immediately back to her left when the view restores to the original vantage point.
Plot hole: In "Aliens of London", it is established on a 'missing person' poster that Rose Tyler is approximately 19 years of age in March 2005 (The poster says 'missing since 6 March 2005'). This means that she must have been born prior to March 1986. However, in "Father's Day", which is dated 7 November 1987, Rose is still a baby in her mother's arms, when she should be a toddler aged approx two.
Factual error: Ida, the science officer, says that the planet is in geostationary orbit around the black hole. However, the word "geostationary" applies very specifically to objects orbiting the planet Earth. Since a black hole is a type of star, this planet's orbit could be described as astrostationary, or even just stationary, but definitely not as geostationary. A science expert on an interstellar mission wouldn't make this mistake, and she wasn't dumbing things down, either, since "geostationary orbit" is already a pretty obscure topic for people unfamiliar with space technology.
Continuity mistake: When the three bride Autons attack Jackie, and they open their hands one at a time, it is clear that when the second Auton opens its hand, all three are already open. After it cuts to Rose, and back to Jackie again, the third one opens (even though we just saw it already open).
Revealing mistake: During the scene in which the Doctor is climbing the transmitter, there is a shot in which he is viewed from above and is then struck by lightning. In this shot, the Doctor's foot disappears briefly as it leaves the green screen on the ground under the actor.
Deliberate mistake: When Jack is scanning the gas mask people, he has a scanner on his right wrist. However, when he shows Rose and the Doctor his image of the Chula warship, the scanner is on his left wrist. The makers flipped the stretch of film, according to the commentary, to make the shot work.
Other mistake: When Miss Hartigan is testing the earpod control, the men start off facing her. She instructs them to turn to the right. They turn 90 degrees clockwise. She then tells them to turn to the left. They turn 180 degrees anticlockwise. However, they should have only turned 90 degrees anticlockwise, and end up facing her again. What they actually do is turn on the spot, as opposed to turning left. It takes a further instruction from Miss Hartigan for them to face her again.
Factual error: When Rose is locked in the room and the bodies come toward her, you can see a modern-day electric light-switch to the side of the door she's trying to get through. When the Doctor hears her screaming for help and runs for her, you can see a central heating radiator. In fact, the undertaker's house has at least two radiators in different hallways. Both are wildly anachronistic for the time period.
Revealing mistake: When Mickey closes the wheelie bin lid, he then finds the plastic stuck to his hands, he lifts his hands up and the plastic stretches like tar. If you look closely when he lifts his hands up for the first time, you'll see that a shadow of one of his hands is cast on his shirt, but there is no shadow of the strands of plastic stuck to his fingers. Additionally, when he turns around to pull away from the bin, he crosses the strands over each other, but they are still parallel in the following scene.