The Unkindest Tut of All - S3-E6
Deliberate mistake: When Batman has figured out King Tut's plot to steal ancient Egyptian scrolls from the Gotham City Library, he and Robin rush away, and in the shot of the library exterior, the words "The New York Public Library" is quite noticeable above the entryway. (00:18:00)
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