Deliberate mistake: When the decoyed Enterprise heads back to Capella, the special effects shot is reversed to indicate that it's going back in the other direction. Unfortunately, this gives us two brief shots of the ship with the registration numbers backwards.
Continuity mistake: When Losira comes for Kirk, he's holding Sulu's tricorder with both hands. When the angle changes to include her, he has it in one hand with his right hand at his side. Cut back to a three shot of Sulu, Kirk and McCoy, and Kirk has the tricorder in both hands again.
Revealing mistake: When McCoy is examining Charlie in sickbay, several shots of the diagnostic panel show a clear reflection of Charlie's upright profile. But Charlie isn't sitting up during the exam - he's lying flat on the table. The close-ups of the panel were shot while actor Robert Walker Jr. was either sitting or standing nearby.
Plot hole: Spock claims that history is unchanged at the end of the episode according to the library tapes - how would he know? If history changed, the tapes would change too. And unlike in "City on the Edge of Forever", there's no Guardian around to keep people from being influenced by an altered timeline.
Continuity mistake: When Kirk and McCoy dispute about the tribbles on the Enterprise bridge, Kirk holds the two tribbles in his hands at alternate heights. In the shots where McCoy's face is seen, he holds them next under his chins, but when Kirk's face is seen, he holds them at chest level.
Continuity mistake: When Sarek enters the banquet room to take his medication, the Tellarite sitting at the table holds a glass in his right hand. In all the full shots with Sarek in the foreground, the glass switches to the Tellarite's left hand. In all the close-ups of him, it's back in his right again.
Plot hole: The crew of the Galileo are stranded because there is a "dampening field" preventing the shuttle craft and its equipment from working. But the device they use to short out the "companion" works.
Revealing mistake: After the ship enters the galactic barrier, when Kirk orders "navigation on automatic" there are flashes of light, and in a closeup of the first console just to the left of the main viewscreen, we can see the special effects material spread across the panel below the controls, for the fire to burn when it explodes. Then it cuts to the wideshot as a second console (near Spock's station) blows up, but in the background that first console is intact - no fire/damage whatsoever (it explodes again in a following wideshot).
Continuity mistake: When Uhura tells Kirk that she's made contact with his private transmitter she's seated at the communications console, but after the contact's been broken when she tells Kirk, "It's impossible at the moment because they stopped broadcasting," in this shot facing Uhura the turbo lift is not behind her - she's at a different station on the bridge - then she's back to where she was once again.
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