Plot hole: When The Penguin is controlling the Batmobile, Batman punches through the floor to take off the transmitter. We still see a video feed of The Penguin. Why? Batman pulled off the transmitter, so there is no reason for there to still be a video feed.
Trivia: Harvey Dent, who would later become Two Face, was originally going to be in this film instead of Max Schreck. The plan was to have Catwoman turn him into Two Face the same way she ended up killing Schreck, except of course Dent obviously would've survived and become Two Face.
[Catwoman is hit by Batman.] Catwoman: How could you?! I'm a woman! Batman: I'm sorry, I... [Batman is hit by Catwoman.] Catwoman: As I was saying, I'm a woman and can't be taken for granted. Life's a bitch, now so am I!