Deliberate mistake: When Batman and Robin get free from the train tracks in the nick of time, the shot of the couple on the passing train is flipped; note things are backwards. (00:04:30)
Visible crew/equipment: At the end Harriet surprises Bruce and Dick with a rehabilitated criminal, and when Blaze walks in we can see the shadow of the moving boom mic on the left side of the screen. (00:23:05)
Continuity mistake: When False Face escapes from the bank he has Blaze with him in the Town Panel truck, and she's handcuffed to the overhead pipe. Blaze is wearing a neutral color nail polish, but in the closeups of her cuffed hand her nail polish is salmon colored. (00:14:30)
Other mistake: While Batman and Robin are pursuing False Face, they stop the Batmobile beside a fence (across from "Bioscope"), and a sign that states "No Trespassing, Desilu Studios" (the show's studio location) is mounted on that fence. (00:15:00)
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