Factual error: A reflection of a full Moon is shown near the end. When Earth is viewed from the Moon earlier, over half of it is illuminated by the Sun. For it to be full Moon, the entire Earth would have to be dark. In addition, the place where the hospital was transported to was dark, and not sunlit.
Continuity mistake: When the Doctor puts his tie back on after returning from the morning, the collar of his shirt is noticeably askew as a result. Just moments later, when Martha enters the TARDIS for the first time, the collar is neat again, without him having done anything.
Continuity mistake: When Martha is talking to the Doctor, who is posing as a patient, about her encounter with him in the street earlier, which he doesn't remember, she takes her stethoscope off her neck. The camera cuts to the Doctor, then back to her, and she's taking it off again.
Continuity mistake: When the Doctor is looking out the window while asking Martha if there's a balcony on the floor of the hospital they're on, in close-ups of him only his right hand is visible, resting on the sill between two windowpanes, and his left arm is by his side. In a wide shot from outside looking in, his left arm is stretched out with his left hand gripping the divider between two windowpanes, in a spot where it should have been visible in the preceding shot.
Factual error: When the Royal Hope Hospital is transported to the Moon, the area it is in is dark, indicating they are on the side of the Moon not currently facing the Sun. However, in a wide shot of the hospital on the Moon, something that looks like the Sun is visible, something that the illumination of the landscape and the Earth shows cannot be visible.
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