Smack in the Middle (2) - S1-E2
Deliberate mistake: Ignoring that its very unlikely all of Jill St. John's copious red hair would fit under a wig that passes for Robin's hair, it's impossible that it would emerge from under that wig looking as full and perfectly styled as it does when she's in the Batcave.
The Purr-Fect Crime (1) - S1-E19
Deliberate mistake: At the end, Batman chooses the button on the right which opens the door on the right, with the tiger inside, but the last shot of the episode has been flipped, so the tiger exits from the door on the left. (00:23:25)
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)
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