The Minstrel's Shakedown (1) - S2-E5
Continuity mistake: When Alfred brings the lunch tray down to the Batcave, Batman is holding a soldering iron, working on the electronic bug. When the shot cuts to a close-up, the soldering iron vanishes, and he's holding a clear plastic probe instead. (00:10:15)
The Minstrel's Shakedown (1) - S2-E5
Plot hole: When Batman and Robin explain to the board members at the stock exchange that Minstrel will not be a threat because they dealt with Penguin, Archer, Clock King, King Tut, and Catwoman, in broadcast order, there is a 6-episode difference between Minstrel and Clock King who has never been seen before. To the first-time viewer, they were probably wondering who Clock King is.
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