Continuity mistake: In 'Becoming, part 1' as Buffy kneels over Kendra's corpse you can see the door over her shoulder, it stays closed all the time. There is a cut to a shot of Buffy from behind showing a cop raising his gun, but he couldn't have come through the door without us seeing him in the last shot.
Continuity mistake: According to Giles in 'Revelations' there are 12 cemeteries in Sunnydale, but in 'Becoming Part 1', amongst others, Buffy agrees to meet someone (Angelus in this case) in 'the cemetery'. If there are 12 in the city, how would they know which one she means?
Plot hole: In the flashback to Buffy in L.A., how did Angel get there with the front windshield of his car blacked out? Being a vampire, he obviously needs it during the day, but he also wouldn't be able to see traffic. And it just seems to be too much of hassle for him to remove it at sunset, just to reapply it at sunrise, just to remove it again, etc.
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