Revealing mistake: In the recap, in the wide shot where Mr Freeze uses his freeze gun on Batman and Robin, you can see they are stunt doubles.
Other mistake: During the final fight scene between the SOCK and POW, Mr. Freeze can be seen in the 76-degree heated section.
Visible crew/equipment: When the photographer is getting a picture of the princess, you can see the shadow of the boom mic on him.
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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