Continuity mistake: The Moldavian host calls attention on the national treasure before the cake is brought. In the wide shot, just before the cut you can spot the man in white uniform turning his head to his right. In the next shot he is turning towards his left. (00:01:05)
Continuity mistake: The cake exploded and the ambassador is on the floor, his back to the tent. In the wide angle shots and in the close-up, the position is quite different; just look at the golden filigree draped over the tent. (00:01:30)
Continuity mistake: When Robin moves away from the window of the art gallery, the shadow he casts is completely different between shots. (00:09:00)
Continuity mistake: When Riddler hands Batman the lawsuit, he is holding the parchment either one or two handed. Robin also goes from being on his knees to standing, head well above Batman's. (00:11:30)
Continuity mistake: Alfred is removing the tray from the table, and Dick leans against the fireplace; between shots, Burt Ward is holding the glass at different heights (look at the mantel decorations). (00:12:45)
Continuity mistake: When Robin has the idea that Riddler placed two other riddles hidden in the legal documents, Robin holds the papers up with his raised right hand, but in the next shot the documents are being held up in Robin's left hand. (00:13:45)
Continuity mistake: Dick waves the lawsuit in front of Bruce telling him it might contain the other 2 clues, prompting a "Great Scott." He is holding it with hist right hand, then the left. (00:13:50)
Continuity mistake: At the secret HQ of the Mole Hill Mob, Molly is wearing a ring on her right hand and many bangle bracelets as she dances, but in the closeup when reaches into the caviar jar the ring and bracelets are gone, then it all reappears. (00:15:00)
Continuity mistake: When Robin parks the Batmobile while Batman heads into the disco, Robin raises the radar antenna on the hood to spin and provide a good signal. But in the shot from behind Robin, as he lifts the dash Batscope cover, the radar antenna is still in its original position and hasn't been raised, but the Batscope visuals are working perfectly. It happens again, when Robin watches on the Batscope as Batman begins to collapse from the spiked OJ, the radar antenna hasn't been raised at all. (00:16:50)
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)
Continuity mistake: While the unconscious Robin is being carried away from the Batmobile by one of Riddler's thugs, his cape loosens and appears to completely fall off just before they exit the frame. In the next shot, Robin is still being carried away but his cape is securely back in place.
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