Continuity mistake: Batman's utility belt is normal in the beginning, but suddenly has a triple sized compartment when Batman goes to get up on the catwalk with his rope. (00:03:30)
Revealing mistake: Right at the end of the obligatory Bat-rope climb scene, look at Robin's cape; the wire that is used to keep it in place (aka sticking out to make it look like it is subjected to gravity and drapes down) loses position too soon, so the real gravity causes the cloth to go towards the wall the dynamic duo is supposedly walking vertically on. (00:11:30)
Continuity mistake: When Batman and Robin scale the warehouse wall then watch Joker and his cohorts, each of the two windows have latches at the bottom, but when the Dynamic Duo burst in through the window, only the left window has a latch. (00:11:45)
Continuity mistake: When Batman and Robin crash through the window, there's no glass left in the frame - at least not until the next shot, when several large shards that weren't there before suddenly appear at the edges. (00:11:50)
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)
Visible crew/equipment: After Joker reveals to his cohorts that his plan involves champagne, when it cuts to Commissioner Gordon standing with Batman at the microphone, as the camera zooms out the chalk T-mark is visible on the floor in front of Batman and another actor's mark in front of Chief O'Hara. (00:18:15)
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: 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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