Other mistake: Riddler says that his "universal wax solvent" can eat through anything. Which begs the question, just what is that vat he cooks the stuff up in made of? We see the wax burn through solid steel vault doors. So it also ought to be eating through the vat, the floor and the bedrock, and sizzling its way down to the Earth's core. (00:07:00)
He Meets His Match, the Grisly Ghoul (2) - S1-E16
Continuity mistake: In the bistro, Batman thwarts Joker's booby-trapped juke box by whipping out his bat shield. Afterward, he places the shield on the floor behind him. In the next shot, as he turns around, the shield has disappeared. He and Robin race off without retrieving it. (00:15:00)
The Penguin's Nest (1) - S2-E27
Revealing mistake: When Bruce, Dick and Aunt Harriet get into what's supposed to be an elevator, there's solid concrete under the sliding doors where there should be a break separating the lift car from the building's floor. It's visible again in Part 2, when Batman and Robin use the same elevator. (00:04:30)
The Clock King Gets Crowned (2) - S2-E12
Continuity mistake: Batman and Robin are trapped in the overturned hourglass. They're sitting upright when Batman suggests an escape plan. In the very next shot, they're sprawled face down on the sand. (00:04:15)
The Devil's Fingers (1) - S2-E15
Visible crew/equipment: When Commissioner Gordon calls Chandell to reassure him that Batman and Robin are back in action, Chandell shares more root beer with Harriet. Then, when he hints for her to leave, a T-mark taped to the floor is visible at the lower left side of the screen. (00:18:20)
Batman Makes the Scenes (2) - S1-E34
Revealing mistake: While Batman and Robin are being held in Penguin's vacuum room, some of the stunt's dark squibs attached to the balloons become visible during this scene. We can even see a squib spark on the last pink balloon. (00:02:15)
The Zodiac Crimes (1) - S2-E37
Visible crew/equipment: Bruce Wayne and Dick Grayson walk into the study alone when Bruce takes the Batphone call from Commissioner Gordon, then just as Bruce says, "The Joker again," at the top of the tuba bell we can see the reflection of a man wearing a white short-sleeved shirt and grey pants. Numerous studio lights are reflected as well. (00:03:15)
The Bird's Last Jest (2) - S2-E28
Visible crew/equipment: At the abandoned Navy rec center, when Penguin and his miscreants are out of bullets they surrender, and as they all gather with their hands up, there are a few chalk T-marks visible on the ground where they stand. (00:05:25)
Hizzonner the Penguin (1) - S2-E17
Other mistake: When Pengy shows up at the Grand Order of Occidental Nighthawks he feigns shock, then as he shoves his umbrella into the vat of sulfuric acid, Penguin moves his arm/hand toward the right to pull out a different umbrella, one that appears destroyed by the acid. (00:22:55)
The Penguin Declines (3) - S2-E39
Revealing mistake: At the Batcave, when Penguin is up on the atomic pile scaffolding Batman grabs hold of him just as Pengy uses his umbrella to float down, and the stunt wire attached to the umbrella ferrule is visible while the duo descend together. (00:21:05)
The Thirteenth Hat (1) - S1-E13
Visible crew/equipment: When Hatter is at Octave Marbot's studio posing as him, Batman and Robin arrive, and chalk T-marks are visible on the floor near Batman's statue in the studio. Then while Batman is in the dressing room, Hatter quietly opens the studio's front door, and more chalk T-marks are visible on the floor in the hall. (00:18:40)
Batman Stands Pat (2) - S1-E14
Visible crew/equipment: When Batman and Robin use the Batmobile's Detect-a-scope to locate Turkey's bowler, as Batman notes that Hatter's hideout must be in the old warehouse, the reflection of the boom pole/mic is visible at the right side of the Batmobile's bubble windshield. (00:16:55)
The Entrancing Dr. Cassandra - S3-E25
Visible crew/equipment: While Batman and Robin are at GCPD, Batman decides to use the library to research occult sciences, and when it cuts to Cassandra and Cabala returning to their hideout they walk over to the table, and behind them we can see the chalk marks on the floor by the door, where they previously stood. (00:05:15)
The Great Train Robbery (2) - S3-E22
Revealing mistake: At the Batcave, when Batman programs the skywriter drone and switches it on, the two stunt wires are visible as it heads out of the Batcave. (00:15:40)
Deliberate mistake: At the start of the episode the Boy Scout asks why the Cossacks are in Gotham City, then a moment later as narrator William Dozier says, "At the Gotham City Museum" the closeup we see shows "Los Angeles County Historical and Art Museum" over the museum doors.
Catwoman's Dressed to Kill - S3-E14
Visible crew/equipment: At GCPD, when Commissioner Gordon wonders how to catch Catwoman before she wreaks even more havoc, at the start of the next shot just as Barbara stands up, the boom mic casts a shadow on the wall as it moves, on the left side of the screen. (00:05:05)
The Foggiest Notion (2) - S3-E12
Continuity mistake: When Batman finds Ye Three Bells at the docks, in the exterior shots the entry door has a handleset on the exterior side of the door, but in the interior shots when the entry door is open, we can see there's a door knob on the exterior side of the entry door. (00:06:25)
The Foggiest Notion (2) - S3-E12
Visible crew/equipment: When Batgirl enters the cricket pavilion at Ffogg Place, two chalk T-marks are visible on the floor between the snuffboxes case and the diamonds case. (00:11:35)
How to Hatch a Dinosaur (2) - S3-E9
Continuity mistake: When Barbara delivers the rare book on paleontology to Professor Dactyl, during this scene Dactyl holds the book with its spine either facing toward the right or the spine facing the opposite way, depending on the angle of the shot. (00:07:00)
Visible crew/equipment: After the first couple of shots of the Batcopter, when it cuts to Mr. Freeze at his refrigerated refuge, the two black tape marks are visible on the floor near Professor Isaacson's and Glacia's feet.
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.
raywest ★