Revealing mistake: Most of the first season's episodes follow a standard outline where, after the opening credits, Batman and Robin arrive in the Batcave via the Batpoles, run and enter the Batmobile and drive off, along the winding, dusty roads toward Gotham, then arrive in front of City Hall, exit the car and run up the stairs. But very quickly it becomes obvious that every bit of this is the same footage used over and over. As they get in the car, it's on a black platform which vanishes as the car starts moving. The same cloud of dust appears as the car's rear heads into the distance on the winding road, and the same crowd of people are in front of City Hall, nearly every episode.
Revealing mistake: The Batcave was a stage set with no real exit for the car. So in many episodes where a shot is looking into the cave from the darker tunnel as the Batmobile approaches and passes to our right, the rear end of the car jumps upward and tire skid marks appear on the floor as the driver slams on the brakes to avoid hitting the studio wall that is out of the shot, while the audio suggests the car is still accelerating.
Deliberate mistake: The Batmobile's Bat-turn lever is located at the center upper console, between the driver and passenger seats. Whenever the Bat-turn lever is used while the Dynamic Duo are in the Batmobile, the same closeup always faces the passenger seat and shows Batman's gloved hand pushing the Bat-turn lever, but the problem is Robin vanishes in this closeup - the passenger seat is empty. Episodes S1xE3 "Fine Feathered Finks", S1xE29 "The Bookworm Turns", and S2xE38 "The Joker's Hard Times" are three examples of the Bat-turn lever being used.
Continuity mistake: When the Batmobile comes in, there's a turntable it parks on to turn it around. Sometimes it's seen for a moment as Batman and Robin get in, but then it's suddenly gone with no evidence of its existence a moment later.
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