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)
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)
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)
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)
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)
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)
Continuity mistake: When Bruce is reading the riddle about vowels, it starts out "We're five little items of an everyday sort." He pushes the button to read the second part of the riddle. Then when Bruce walks around, the first part of the riddle can be seen again. (01:33:25)
Continuity mistake: When Bruce wakes up after getting shot in the head by Two-Face, there is a "BW" (Bruce Wayne) on his pillow next to his left ear. But after Alfred asks, "How are you, young man?", the "BW" is gone. (01:35:55)
Continuity mistake: After the Bat-Boat is destroyed, it changes to a normal wooden boat. (01:41:50)
Continuity mistake: At the end of the film right after Batman has saved Chase and Robin from falling and Two-Face appears on the ledge, you can see that in one shot Chase has lost her shoes and in the next shot she's wearing black sneakers. (01:48:00)
Continuity mistake: Right before The Riddler is about to kill Robin and Dr. Chase, his finger changes position on the button on his cane between shots. In one shot it is next to the button but in the other it is suddenly completely covering the button. (01:57:15)
Answer: It's called "Smash it up" by The Offspring.