The Bird's Last Jest (2) - S2-E28
Continuity mistake: When the Dynamic Duo goes up the Bat Pole chute, the NEGATE COSTUME CHANGE lever is down. When they come down as Dick and Bruce, the lever is up, and Robin pulls it down to stop the instant change back to Dick and Bruce.
The Clock King's Crazy Crimes (1) - S2-E11
Continuity mistake: After starting the sand in the giant hourglass, Clock King puts on his alarm clock top hat. Between distance and close shots, the clock reads a few minutes to three, then Three o'clock in the closeups.
Not Yet, He Ain't (2) - S1-E22
Continuity mistake: When the Batmobile is blocked by the second police car, it's stopped on an angle, and almost touching the car. When the Penguin commandeers the Batmobile, it's sitting correctly on the road, so that it can be driven easily without really needing to back up.
The Joker Goes to School (1) - S1-E15
Continuity mistake: As Batman goes to cuff the Joker, the Joker holds up his right hand. In the close up, his left hand is raised.
Continuity mistake: Batman and Robin are being summoned by Commissioner Gordon, but are still in civilian clothes, so it's time to get to the Bat-cave. Bruce presses the switch and the bookshelf begins to slide open. Pay attention to the white books in the third row and the red books on the top shelf. In the close-up that follows there are fewer white books, their cluster aligned with the last of the red books, a different proportion from the previous shot. The name tags on the bat-poles are also misaligned in the wider shot and perfectly at the same level in the close-up (which is a reused shot from the pilot episode). (00:05:10)
Continuity mistake: Despite his lackadaisical pitching, the Joker manages to eliminate a series of batters. In particular, there is one elimination before and after the warden tells O'Hara about the joys in life, when Joker didn't really have time to throw the ball 2 more times. (00:01:55)
Continuity mistake: The Joker sends a doll crashing through the window of Commissioner Gordon's office. O'Hara is holding in close-up the doll in a different way from the wider shots, and the jester cap of the figurine is also blatantly folded in a different fashion. (00:18:30)
Smack in the Middle (2) - S1-E2
Continuity mistake: After the Garbo joke, the various extras begin to pass out from the gas with multiple inconsistencies between shots. In the shot that follows the joke, the woman with the red and black dress is fainting, but she's back up in the next, in which the policeman goes down, only to be up in the one later, etc. (00:19:35)
Smack in the Middle (2) - S1-E2
Continuity mistake: Riddler wakes Robin up, and the Boy Wonder does not waste time in his tongue lashing. During the scene, the lighting on set is very noticeably different; with the camera positioned between the two, Frank Gorshin's jaw casts a shadow; with the camera behind Burt Ward, it's him casting a big shadow on the green unitard. (00:03:40)
The Penguin's a Jinx (2) - S1-E4
Continuity mistake: At the beginning of the big fight with Penguin, everyone is doing some fencing with the umbrellas. The guy with the caveman face on the left switches stance depending on the camera angle; he's lunging forward waving his umbrella with one arm in the frontal views, and he's in a guarded position in the overhead shots. (00:21:20)
Continuity mistake: Arriving at the museum, the dynamic duo is greeted by some rabid Robin fans; when they start screaming and cheering for him in a close-up, there are men passing by behind them that were not there in the wider shot with the Batmobile approaching. (00:09:40)
The Penguin's a Jinx (2) - S1-E4
Continuity mistake: Robin jumps on a table, dodges a swing and gives a (really weak) kick to his adversary. Cut, and he is nearly tripping over two umbrellas on the table he is standing on, suddenly appeared out of nowhere. (00:21:40)
Fine Feathered Finks (1) - S1-E3
Continuity mistake: The bat-phone rang, and Dick is practicing his French. Bruce in the wide shot has taken his pen off the sheet, but he is in a different position in the close-up that follows. Same when he mentions the "key to world peace"; his hand changes position. (00:03:25)
Continuity mistake: When Joker shows off for the first time his utility belt, it switches sides between shots. First each pocket has its buckle on the right, towards the blonde bimbo, then in close-up the buckles are on the left, and when it switches back to a wider angle, again they are on the right. (00:16:45)
Smack in the Middle (2) - S1-E2
Continuity mistake: Riddler sedates again Robin with the "B" spray. He puts his own mask back on, and on the table the spray can is turned with the letter facing away from the camera. Riddler takes a peek at the unconscious Robin, and in the next shot both sprays have their letters properly showing, only to be hidden again when the same camera angle as before is used again. (00:05:55)
Fine Feathered Finks (1) - S1-E3
Continuity mistake: The dynamic duo enters the umbrella factory and begins talking with the Penguin. Right away you can see the cigarette in the holder has its tip turned into ash, but in the close-up on Burgess Meredith's grin a greater part of the cigarette is consumed. (00:16:40)
The Penguin's a Jinx (2) - S1-E4
Continuity mistake: Batman punches Hawkeye with a BAM; in the closer view Robin is not where he is supposed to be (next to the elevator, where he was sent reeling by the punch and where the action takes place in the following shot). (00:21:45)
Continuity mistake: The first fight, in the museum, besides the obvious dose of wild haymakers has a rather messy phase when the "THWAPP!"; appears; Robin is facing two enemies, but the one with the red shirt lost his hat, which is back on in the next shot (and was on in the previous). (00:14:10)
The Penguin's a Jinx (2) - S1-E4
Continuity mistake: Batman sends Swoop into the wall with an "OOOOFF!" onomatopoeia. The impact knocks down from the shelf a bunch of umbrellas that fall on the poor guy, but as the scene goes on, the floor is clean and the umbrellas are still on the wall rack. (00:21:30)
Continuity mistake: During the climactic fight, Batman tosses the Joker utility belt, causing confetti and streamers to fall down all over the place in some shots, while there is no such paper rain in others. (00:21:40)
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