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)
Factual error: A fictional country is shown in this episode, with the name "Republic of Moldavia." It is portrayed as a south-eastern Asian country (the exterior shot shown was from the Thailand exhibition at the 1964 New York World Fair), but it's quite an odd choice to use as fictional name a name of a principality that historically did exist, in a completely different part of the world, where the current Moldova is.
Smack in the Middle (2) - S1-E2
Revealing mistake: Of all the possible mistakes related to the Robin mask Molly supposedly wears, the one that really could have been avoided was the fact that the camera does not cut away from Burt Ward while he is in the middle of the gesture of turning around and 'ripping away the mask', but only after he's done it, so you can tell nothing came off, nothing moves in his hair, let alone the face itself. (00:10:25)
The Penguin's a Jinx (2) - S1-E4
Other mistake: Penguin says that his sleeping gas' effect lasts 18 hours, but Bruce in his shop woke after mere minutes (sure, the extreme temperature helped him regain consciousness, but something as potent to knock you out for 18 hours with a small whiff can't completely lose effectiveness just because your feet are getting a little hot). (00:20:15)
The Penguin's a Jinx (2) - S1-E4
Plot hole: Penguin's magnet attracts with amazing force Batman and Robin's utility belts, to the point of pulling them in from the other side of the room; it's however ludicrous to think that no other metal object would be in the apartment and fly at a magnet with that sort of power. (00:14:30)
The Penguin's a Jinx (2) - S1-E4
Revealing mistake: When Batman throws the first punch at Burgess Meredith, he is hitting him with an uppercut so wide that it misses by a mile even the tip of his cigarette holder, it's incredibly fake even for the understandably low and cartoonish standards of this show. (00:21:30)
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)
Revealing mistake: The newsreel shows several issues of the imaginary Gotham City Times. Nice job with the headlines and the page layout, but every single one of them is issue number 11. (00:14:00)
Fine Feathered Finks (1) - S1-E3
Continuity mistake: When Batman opens his drawer full of electronic (not to mention, literal) bugs, the various pseudo-creatures are arranged in different ways between the wider angles and the close-up. (00:20:50)
The Penguin's a Jinx (2) - S1-E4
Continuity mistake: When the umbrella fight erupts, the red suitcase with the money changes position on the corner of the counter (basically rotating of 90°). (00:21:20)
Fine Feathered Finks (1) - S1-E3
Continuity mistake: The tiny spider microphone rests on the back of Robin's right hand, but the close-up shows a gloved left fist. (00:20:55)
Fine Feathered Finks (1) - S1-E3
Continuity mistake: Adam West's hands inside the net are in a different position when Penguin sprays him with the red gas. (00:22:20)
Continuity mistake: In the cliffhanger, Riddler is putting a vice around Robin's head and cranks it; in the close-ups he turns the mechanism between his fingers, while in the shot from behind Robin's head he is turning a little handle. The height the vice is placed at is also different. (00:24:20)
Smack in the Middle (2) - S1-E2
Continuity mistake: Batman, after losing his breather between an "Urkkk!" and a "Zok!" chases down a goon that is making a run for it going for the window. Two other minions are following Batman, but their motion repeats in the next shot, when it's "Zzzzzwap!" time. (00:22:00)
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