Continuity mistake: The Doctor pulls the arm off Rose, deactivates it, and the fourth and fifth fingers are the last to stop moving. When the angle changes, the first two fingers are the last to stop.
Audio problem: When Rose is sitting in the observation deck, after she looks at the tree cutting and suggests they might be related, when she puts it back down, it makes a "clink" sound, as if made of metal, despite being in a sort of cardboard cup. (00:15:20)
Continuity mistake: When Rose pulls out her phone and checks for a signal, she's holding it with two hands in the shots facing her, but only with one hand in the shots from behind her. (00:17:50)
Continuity mistake: When the Doctor is putting a new battery in Rose's phone, which hand he's holding it with keeps changing between shots. (00:18:00)
Continuity mistake: As the room that Rose is in gets scorched by the blazing sun, the level of burning is above her head in close up shots, but at Rose's head height in wide shots. (00:29:40 - 00:30:01)
Continuity mistake: When the Doctor is standing in front of the giant whirring fanblades, just before he passes the first one, which side of his jacket is blowing away from his body changes between the wide shot and the close-up. (00:35:20)
Continuity mistake: Rose Tyler's hairstyle changes from shot to shot when she is discussing being 'the last human' with Cassandra. Rose's hair is in one style for the long shots, and another style for the close-ups.
Continuity mistake: Early on, when Rose is talking to the Doctor about how amazing it is that they can visit days that only happened once, her hair moves from tucked behind her ear to covering it, then back to its first position. (00:05:00)
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. (00:16:30 - 00:17:08)
Suggested correction: While the light switch is ahead of its time, the heating radiator was invented by Franz San Galli in 1855 and was immediately popular. It's possible that by 1869, when the episode takes place, that Mr. Sneed would have one.
Other mistake: At the end, the Doctor and Rose treat Charles Dickens by making the TARDIS dematerialise in front of him. The engines begin to grind, but the time rotor (the thing in the console's central column that goes up and down when the TARDIS is flying) does not move. (00:42:53)
Continuity mistake: When Jackie is yelling at the Doctor about Rose's yearlong disappearance, in a shot facing her she turns her head to her right, then in a shot facing the Doctor she does it again. (00:04:00)
Continuity mistake: When the Doctor and Rose are talking on the roof, just before the spaceship appears, Rose's hair is blowing in the wind in shots facing the Doctor, but mostly still in shots facing her. (00:05:10)
Revealing mistake: When the alien spacecraft crashes into Big Ben, it is obvious that the shot has been reversed/flipped around. Look at the Roman numerals on Big Ben's clock face: They should read (clockwise from the top) XII, I, II, III, and so. Instead they read IIX, IX, X, etc. (00:06:35)
Visible crew/equipment: During the news broadcast on the disappearance of the Prime Minister, shown on an in-universe TV screen, the reflection of the camera filming the TV is visible. (00:12:00)
Continuity mistake: During Harriet Jones' introduction, her grip on her ID card changes between shots. (00:12:25)
Continuity mistake: When Indra Ganesh offers Joseph Green the briefcase with the Emergency Protocols, he's holding it with the handle in his right hand in wide shots, but with it in his left hand in close-ups of him. (00:12:36)
Continuity mistake: When the coroner is interrupted by a banging coming from the freezer while trying to do paperwork, she's shown turning around and standing up twice, in a close-up and the long shot. (00:19:20)
Revealing mistake: In close-ups of the modified pig, seams are visible on its forelimbs. (00:22:45)
Continuity mistake: When Jackie is reporting the Doctor to the alien helpline, her grip on the cordless phone changes between shots. (00:29:55)
