Continuity mistake: At the very end of the episode, when Batman, Robin, and Batgirl find out Joker is released from prison, one shot shows Joker in his usual pink suit. But in the very next shot, his suit is gray.
Deliberate mistake: At the start of the episode the Boy Scout asks why the Cossacks are in Gotham City, then a moment later as narrator William Dozier says, "At the Gotham City Museum" the closeup we see shows "Los Angeles County Historical and Art Museum" over the museum doors.
Visible crew/equipment: When Olga shouts, "Da, da, da! Drowning her in caviar!" in the overhead shot, the boom mic repeatedly dips into view over Olga, at the top of the screen. (00:21:55)
Visible crew/equipment: After Egghead dumps Batgirl into the vat of caviar, Egghead, Olga, and the Cossacks attempt to escape, then just as they run out the door, a bit of set lighting equipment (which wasn't in previous shots) is visible at the left side of the screen. (00:22:10)
Answer: There wasn't one singular city used. It was shot in and around various locations in California, plus on a lot of sets and backlots. Ex. Some scenes were shot in LA, some in Pasadena, some in Santa Barbara, etc.
TedStixon