Factual error: Edith Keeler says to McCoy, "My young man is taking me to see a Clark Gable movie." In 1930, Clark Gable was an uncredited bit player who had last made a film in 1926 - Edith, a New York charity worker, would not even have encountered his name.
Continuity mistake: As Kirk and Spock prepare to enter the time portal, Spock deliberately places the tricorder at his left side, holding it there with his left hand. As they appear in 1930 earth, he is holding the tricorder with his right hand and his left hand is free.
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.
Continuity mistake: After extinguishing the fire, Spock pulls a large vacuum tube out of the primitive computer power supply he's constructed on the bed. When the shot changes, the tube is back in place again.
Other mistake: When talking to Edith Keeler one evening, Kirk refers to a star in Orion's belt, pointing it out. The shot then cuts to a scene of the horizon. The sky is quite cloudy and the light from the sun is still lighting the horizon. The stars would not be visible.
Continuity mistake: The bundle of clothes Kirk steals from the balcony changes several times while he and Spock face the policeman. The red shirt is not visible, then is hanging down on the right, then isn't. It then disappears from view until they reach Edith's basement, where Kirk puts it on. Two belts, two jackets and Spock's cap also seem to have added themselves to the bundle somehow. They're not there when Kirk first dismantles the tangle of clothes in the basement.
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