Continuity mistake: After Gary turns on the typewriter and tells Roberta to give her report but not with her fingers, she responds, "Well, how do you expect me to type, with my nose?" But the typewriter types, "How do you expect me to type, with my nose?" omitting the "Well." It didn't edit her question, because we see from the rest of the scene that it types absolutely every word she says until Gary turns it off. (00:17:45)
Continuity mistake: Kirk is talking to Scotty on his monitor, after they have finished talking Kirk goes to switch off the monitor by pressing the button just below it. If you watch very closely you will notice Mr Scotts image fades from the screen before Kirk even touches the button. (00:07:05)
Plot hole: Scotty is receiving images of the launch, supposedly from an orbiting satellite. But many of the camera views are obviously from ground level, and couldn't possibly have been taken from orbit. It's not a rebroadcast from a ground camera, either. The implication is clearly that Scotty is is picking up a live satellite feed. (00:34:30)
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.
Revealing mistake: After Scotty finally locates Seven on the gantry while he's tampering with the rocket, it cuts to a quick shot of Kirk and Spock, then in the following shot of Roberta when she turns the vault's combination locks, the stunt wire is visible behind her as it moves up to turn the door's wheel handle. (00:35:25)
Continuity mistake: To open the secret doors, Seven tilts one of the pen holders down to the top of the desk. After he's gone, Ms. Lincoln comes in looking for him, and the pen is upright without it having been pulled back up, unless it was supposed to reset itself.
Continuity mistake: As Kirk talks to Chekov on the screen, his hands are flat on the table, and there is a collection of data tapes in front of him. After the angle change, his hands are clasped on the table, the tapes have been replaced by a data pad, and Kirk appears to be frozen.