Continuity mistake: Towards the end of the episode when the Enterprise is moving leaving from the starbase, the ship is at an angle to the camera pointing to the right of the screen. The starbase behind it gets smaller and goes up a little. If the Enterprise were leaving the starbase at that angle, the starbase should have moved to the left and then completely off the screen.
Continuity mistake: During McCoy's struggle with the orderly, only half of the plastic colored bottles strewn on the table are knocked off onto the floor. But when Nurse Chapel hurries over to tranquilize the orderly, the rest of the bottles vanish, leaving the table completely clear. (00:19:10)
Continuity mistake: Mr. Leslie appears as one of the guards in the space station, Stardate 45XX.X. Unfortunately, he was killed in 'Obsession,' Stardate 36XX.X. External websites have a variety of excuses for this miraculous reappearance, but he very definitely dies in the show, with no canonical reason for him reappearing beyond it being a mistake.
Continuity mistake: In a few episode we'll hear about "Hodgkin's Law of Parallel Development," but here Spock claims the odds of a planet developing similar to Earth's Nazi culture are "virtually impossible." Why was it so likely as to be easily accepted there, but impossible here?
Continuity mistake: After Edith stumbles and nearly falls down the stairs, she still has both her shoes on. But after the kiss, she proceeds up the stairs and is now carrying one shoe. She had no time to take it off, as she was, er, busy with Kirk the whole time. (00:43:55)
Continuity mistake: In the very first scene after the intro, when Bela's goons hold the landing party at gunpoint, Spock's position on the sidewalk shifts back and forth from standing on the other side of a wooden bench to standing alongside Kirk and McCoy with no bench in between. In fact, in the latter shots, the bench has moved all the way across the street. (00:04:50)
Continuity mistake: At the end, when Kirk and Garrovick are beamed up seconds before the explosion, Kirk is holding his communicator up to his mouth with both hands. But when they finally materialize on the Enterprise, Kirk's hands are down at his sides: no communicator in sight. (00:47:00 - 00:48:10)
Continuity mistake: The aging Kirk's hairline has receded considerably. When he ages still more and goes completely gray, however, the receding process inexplicably reverses itself, and he suddenly has much more hair on his forehead than before. (00:34:35 - 00:43:55)