Continuity mistake: When Batman and Robin are in the museum speaking with the director, Batman's chest emblem is high enough on his chest that it nearly touches the bottom edge of his cowl. In the next shot, when he leaves the museum, the emblem has moved down several inches.
Continuity mistake: In close-up, Bruce Wayne folds his newspaper in half, leaving the folded edge facing the camera. The shot cuts to a wider view and immediately, the paper has reversed positions in his hand. (00:04:00)
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)
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)
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)
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)
Continuity mistake: Everytime the Joker is shown on TV his hair is brown, but before and after the broadcasts, it's his usual green with no time for him to change it.
Answer: It's wasn't unusual for multiple actors to play one character on this show. The villains on the TV Batman were played by guest stars, not regular cast members. George Sanders, Otto Preminger, and Eli Wallach played the part at various times. They may simply have only wanted to play the part once or twice, or they were later tied up with other projects, making it necessary to cast someone else. Other characters, like Catwoman, were also played by more than one actress.
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