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. (00:02:45)
Character mistake: In an early scene with Yvonne Hartman, she says "We've just measured the ghosts' energy at 5000 gigawatts." There are at least two mistakes here: (1) watts are a unit of power, not of energy (that would be joules); (2) later on, when Hartman is showing the Magnaclamp to the Doctor, she refers to imperial tons and says "Torchwood refuses to go metric", but a gigawatt is a metric unit.
Continuity mistake: When Yvonne Hartman is showing the Doctor through Torchwood after leaving the TARDIS, she goes to push a door open into the main building with one hand, but when the camera cuts to them entering the building she pushes them open with two hands. (00:20:10)
Continuity mistake: When Yvonne is taking the Doctor and Jackie to the Sphere Room, there is a sign opposite the door that has the words "Torchwood Institute". In the next shot where you can see the corridor from inside the Sphere Chamber, the sign has disappeared. It reappears again when Rose is going into the Sphere Chamber, but this time it has directions to the Reception, Canteen and Lever Room as well as "Torchwood Institute". It also disappears again when you see the corridor from inside the Chamber. (00:23:30 - 00:31:45)
Continuity mistake: When the Sphere Chamber is first shown, there is a rolling staircase set up just below the Sphere. Rajesh Singh is shown climbing the staircase and nearly touching the Sphere. However, when the room is next seen, when Raj hears that the Doctor is coming and walks up to look at the Sphere again, the staircase has vanished. Later, when the Doctor and Jackie are shown into the room while Yvonne gives them the Torchwood tour, the staircase is still missing, and it's more obvious that it is no longer in the room. It does not reappear for the remainder of the two-parter.
Continuity mistake: When the Doctor uses his sonic screwdriver on the glass partition in the Torchwood Institute, cracks radiate out in a star pattern, meaning the glass will break into pointed shards. When it does break, it breaks into cubes, like safety glass. (00:29:30)