It's the Way You Play the Game (2) - S2-E26
Visible crew/equipment: After Batman gets out of the Batmobile, you can see the lighting rigs, boom mic, camera and crew reflected on the Batmobile's side.
Flop Goes the Joker (2) - S2-E58
Revealing mistake: At the "Batfight", the Joker leaves the room and he kidnaps Baby Jane but when it cuts back to the fight, if you look closely at the door, the Joker can be seen leaving the room again. Note his iconic pink suit.
Caught in the Spider's Den (2) - S2-E56
Visible crew/equipment: When Batman and Robin are caught in Black Widow's web, the wires used to operate the spiders are visible.
A Riddling Controversy (2) - S2-E46
Visible crew/equipment: During the "Batfight" at the end, it is extremely obvious that the Riddler is a stuntman and not John Astin, note that the stuntman does not have a mustache.
Other mistake: At the end of the episode when the cattle are stampeding, the sign over the Saloon is backwards.
Hizzonner the Penguin (1) - S2-E17
Continuity mistake: The indicator arrow alternates between AGONY and KO too quickly.
Hizzonner the Penguin (1) - S2-E17
Continuity mistake: The man with the tambourine goes from standing still just shaking the tambourine to moving wildly between shots.
Penguin's Disastrous End (3) - S2-E44
Visible crew/equipment: As O'Hara goes to move his patrol car, the boom mic and lighting equipment is reflected on the windows.
The Joker's Hard Times (2) - S2-E38
Continuity mistake: When the Joker directs Gordon and O'Hara to look at the statue of Justice, they look at it from a front view. But when Joker comes out of the Police building where he left them frozen, it's from behind the statue, rather than in front.
Pop Goes the Joker (1) - S2-E57
Other mistake: Not only can Caesar Romero's moustache can be seen, but the line between the white makeup and his skin on his neck. Also, when Robin's face came out of the clay, his mask was almost around his mouth. A moment later, the mask is back in place. Later, his face, mask, and body is spotless.
Flop Goes the Joker (2) - S2-E58
Revealing mistake: After Bruce stops the governor of the mobile, it can be seen that the weights are painted Styrofoam balls.
Continuity mistake: At the end, when Minstrel is cornered by the Dynamic Duo, he pulls out his sparking remote unit. After being told it won't work, as O'Hara cuffs him, the unit that was in his hand disappears.
Dizzonner the Penguin (2) - S2-E18
Other mistake: In the fight in Convention Hall, when the goons surround Batman, watch the one on the front left. He's punching over and down, yet he's hitting nothing.
Dizzonner the Penguin (2) - S2-E18
Other mistake: Batman throws himself into acid. Counting on acid-proof suits, Batman comes out unscathed, but his ropes didn't get burned off until he dipped the back into the acid. Also, even though he covered his face a moment, he came out with his face exposed, and undamaged, when it should have been burned.
Dizzonner the Penguin (2) - S2-E18
Continuity mistake: In the beginning, Batman and Robin's side of the scales are lowering them down. At one point, it isn't moving.
Hizzonner the Penguin (1) - S2-E17
Continuity mistake: At the end of the episode, the pointer on the scales alternates from pointing between "AGONY" and "K.O." to pointing between "CLOSE" and "AGONY" and back again. (00:22:05)
A Riddling Controversy (2) - S2-E46
Continuity mistake: In the park at the statue base, Batman uses a knife to open the telegram. As he reads, the knife is pointed up, and a moment later, it's down.
A Riddling Controversy (2) - S2-E46
Revealing mistake: When Batman and Robin use their jets to get out of the quicksand, as they fly up, you can see the wires used to pull them up.
The Joker's Provokers (2) - S2-E22
Revealing mistake: When Alfred talks to his twin Egbert, as they face each other, you can plainly see the line dividing the screen so that Alan Napier can play both parts.
An Egg Grows in Gotham (1) - S2-E13
Revealing mistake: A lot of the time, especially in close ups, you can see the line between the bald dome wig and Vincent Price's real skin, as well as the different coloring of skin.
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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