Pop Goes the Joker (1) - S2-E57
Continuity mistake: After Oliver Muzzy congratulates Joker for his magnificent artwork and makes a deal with him, before Joker leaves the gallery he runs his finger across Batman's yellow emblem to sign his red artwork, but at GCPD HQ when they're discussing the matter, Joker's signature is completely different, it's now a capital "J" on Batman's yellow emblem. (00:06:45)
Pop Goes the Joker (1) - S2-E57
Other mistake: Not only can Caesar Romero's moustache can be seen, but the line between the white makeup and his skin on his neck. Also, when Robin's face came out of the clay, his mask was almost around his mouth. A moment later, the mask is back in place. Later, his face, mask, and body is spotless.
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