The Penguin's a Jinx (2) - S1-E4
Plot hole: Penguin's magnet attracts with amazing force Batman and Robin's utility belts, to the point of pulling them in from the other side of the room; it's however ludicrous to think that no other metal object would be in the apartment and fly at a magnet with that sort of power. (00:14:30)
The Minstrel's Shakedown (1) - S2-E5
Plot hole: When Batman and Robin explain to the board members at the stock exchange that Minstrel will not be a threat because they dealt with Penguin, Archer, Clock King, King Tut, and Catwoman, in broadcast order, there is a 6-episode difference between Minstrel and Clock King who has never been seen before. To the first-time viewer, they were probably wondering who Clock King is.
Enter Batgirl, Exit Penguin - S3-E1
Plot hole: When he abducts Alfred, believing him to be a minister, Penguin suffers a peculiar memory lapse. He's encountered Alfred twice before in previous episodes, even kidnapped him once before (knowing full well that he was Bruce Wayne's butler). Yet here, he fails to recognize Alfred at all. (00:12:30)
I'll Be a Mummy's Uncle - S3-E23
Plot hole: After the rock falls on King Tut at the end giving him amnesia, Batman explains how the rock fell. He says Tut raised his voice three decibels above high C. Decibels are a measure of volume, while pitch is a measure of frequency: the two are not directly related. This mistake was done in other episodes of Batman also, such as when Joan Collins played the villain.
The Joker's Flying Saucer - S3-E24
Plot hole: A time bomb is placed under the dashboard of the Batmobile and goes off inside the batcave. Much of the batcave is ruined, including fallen I-Beams, but the Batmobile has no damage to it even though the bomb was planted in it.
Answer: There wasn't one singular city used. It was shot in and around various locations in California, plus on a lot of sets and backlots. Ex. Some scenes were shot in LA, some in Pasadena, some in Santa Barbara, etc.
TedStixon