Continuity mistake: In the vault, just before Batman enters, we get a nice shot of the bank guard. He appears to be lying length-wise in the vault, his body flat and straight, and his head laying on the "step" toward the left side of the screen. When the scene cuts to Batman entering the vault, the guard is positioned differently, at much more of an angle, and his head is higher, closer to the second row of boxes from the bottom of the vault. (00:07:25)
Audio problem: When boiling acid starts spraying inside of the bank vault that Batman and a security guard are in, the security guard shouts "It's boiling acid." But his lips does not match with what he is saying. It happens again when the guard yells to Batman, "Don't let go." (00:08:15)
Continuity mistake: While retrieving the safe from the chopper, Batman tells the bank guard to "hang on." The guard holds on to the chain to which the safe is attached. But as the safe comes back into the vault, the bank guard is holding on to one of the tubular structures of the safe itself. (00:09:35)
Continuity mistake: When Nygma is reading his boss' mind he pulls the lever to add more power. The way he holds it differs between shots. (00:20:22)
Plot hole: In the scene at Wayne Manor where they show Harvey "Twoface" getting scarred with the acid on TV, you can see the acid cover almost his entire face before he puts the folder over to protect it. (00:24:15)
Continuity mistake: When checking a tape that shows the (fake) suicide of Stickley, Bruce is handed the suicide letter, which was folded in three parts. The next shot shows a clean sheet of paper, with no wrinkles or folds. (00:26:20)
Continuity mistake: During Bruce's first session with Dr. Chase, when the two are talking about the dream doll, she puts a jacket on. The way the hood lies on her left shoulder changes in every shot. Sometimes, the white part covers her shoulder, sometimes it doesn't. (00:31:20)
Continuity mistake: In the scene at the circus when the ringmaster announces "The Flying Graysons", Robin's mom is standing on his left side. In the very next shot, she is on his right side. Also, when Robin is doing the "Death Drop", his mom is on his left and his brother is on his right side. After he completes the stunt and he swings back up with his dad, his mom and brother have switched places. (00:32:45)
Continuity mistake: After causing Two-Face to accidentally blow up a car that had a couple members of his gang in it, with his missile-launcher, Batman speeds by Two-Face with the Batmobile as he is being chased by other members of Two-Face's gang. In one shot the Batmobile is very close to Two-Face, but in the next shot the Batmobile is far away from Two-Face (when he tries to throw his missile-launcher at the Batmobile). (00:49:05)
Visible crew/equipment: To escape Two-Face's ambush, Batman tilts the batmobile onto its rear wheels, and thanks to the bat-hook, he's able to use the vertical facade of a building as an alternate route. As the batmobile rides up the building, if you look closely you can see what looks to be a wire behind the batmobile (probably helping the batmobile or a model of it travel smoothly up the building or a model of the building). (00:50:35)
Continuity mistake: When Chris O'Donnell does the laundry and is putting wet clothes on the string, he takes a clothing item and wraps it around his right leg then twists water out of it. When he does this his right foot goes in the air a few inches above the ground but in a closeup to his foot and water falling it is his right foot standing when it should be his left. (00:55:00)
Continuity mistake: When the Riddler and Two-Face are in their getaway van after robbing the jewelery store, the Riddler has his green hat on, but in the next shot, it has switched to a tiara. (00:55:40)
Continuity mistake: When Bruce is at Dr. Meridians place, Bruce receives a call from Alfred. Bruce moves his finger on the side of the communication device up. The next shot shows him doing it again. (01:02:05)
Continuity mistake: In the scene where Robin steals the Batmobile, the first shot of his joy-ride shows twin fins on the rear of the batmobile on 45 degree angles, in the following shots they are all standing directly vertical. (01:02:40)
Character mistake: When Bruce is looking at Chase's various clippings of Batman on her desk while she's tending to the tea kettle, there is a paper that says "Diagnosis Form #39," and just above it, the word "schizophrenic" is incorrectly spelled as either "schizofrenic" or "schizoprenic" (a metal clip is partially covering the last six letters of the word, so it's hard to determine whether it's an "f" or a lone "p" where the "ph" should be). (01:03:11 - 01:03:32)
Continuity mistake: At Edward Nygma's party at the Ritz Gotham there is a fountain with a large version of his "box" on top. When Two-Face comes, he shoots the "box" and destroys it. There should be some parts left over but when Batman falls through the ceiling he lands on the fountain and there's no remains of the "box." (01:12:15 - 01:13:30)
Continuity mistake: In the part after The Riddler shows Two-Face the newspaper with the headline "Batman Survives Subway Sabotage," The Riddler pushes a glowing green cylinder into the monkey's mouth and the camera cuts to the screen so they can see Bruce Wayne's brain. When it cuts back to Two-Face and The Riddler, the cylinder isn't there. (01:25:54)
Revealing mistake: When Two Face and Riddler enter Wayne Manor, and knock Alfred down. Alfred, unconscious, pushes the dinner tray aside, so the thugs can carry him into the closet. (01:27:55)
Continuity mistake: In the dream/memory where young Bruce is falling into the cave, the red book is in his right hand as he starts to fall. It is dark and raining. The next shot shows him at the top of the hole falling and he has dropped the red book. It is also bright outside with no signs of rain. The next shot shows him with the red book in his left hand. When he stands the book is in front of him, too far away if he held it. (01:30:00)
Continuity mistake: When Bruce slides across the table, both pumpkins go the right near Dr. Chase. In the next shot, they are on each side of the table. (01:32:55)
Answer: It's called "Smash it up" by The Offspring.