Continuity mistake: Buffy's birthdate starts at 1979. In "I Robot, You Jane", when the demon pulls up Buffy's file in the school system, it shows her year of birth as 1980 and her year as sophomore. But when her file comes up on the student's computer screen, it's changed to 1979 and that she is a senior. It wasn't until towards the end of season 1 in "Nightmares" that her birth year is 1981 (on her tombstone before she comes out as a vampire).
Continuity mistake: When Buffy is talking to Owen at the end, there is a woman wearing a red/black checkered jacket, sometimes carrying a red bag, walking up the stairs behind Owen several times. You can also see her behind Buffy in shots facing her.
Continuity mistake: When Buffy finds Dr. Gregory's glasses on the lab floor she places them on the table, from the long shot she places them at the left end of the table near a plastic tub with a white lid, with the glasses open. Then it cuts to a close up where they are in front of a book on the right side of the table and are closed. Then cut back to the long shot and they are in the original place.