Continuity mistake: When the Joker's henchmen are trashing the museum, one guy puts red handprints all over a portrait, later the Joker walks by the same portrait and the handprints are gone. (00:59:55)
Continuity mistake: After Batman rescues Vicky he has a voice remote for the Batmobile. As the car approaches he orders the car to stop by speaking into the remote and then lowers it to his side. When the camera shot changes showing the car stopping right in front of him, he is still holding the remote to his mouth. (01:09:20)
Continuity mistake: When Joker is at the top of the building and taunting them, his chin is red (presumably from his blood when Batman socked him in the jaw.) After he falls off the building, and we see him lying on the ground, his chin is not covered in blood like it was before. (01:50:45 - 01:54:35)
Revealing mistake: Harvey Dent and the mayor argue in Dent's office which has an unusual semicircular shape. In the next scene, Alfred uses the telephone in Wayne Manor, and you can see that it's the same room, just with different furniture and dressings. (00:56:10)
Continuity mistake: When Batman initially fires the rope around Joker's ankle at the end, the rope only goes around the statue's legs. The next time we see it, it's somehow re-tied itself, and is now looped around the statue's backside. (01:52:55)
Visible crew/equipment: When Batman comes through the skylights in the museum just after Vicki throws water on Joker's face, you can see the stunt cable that lowers Batman down. Batman does use a system of cables and grappling hooks because he has no superpowers, but the cable lowering him down disappears as soon as he hits the ground. It was used to lower the actor down safely, not the character.
Continuity mistake: When Bruce smashes the vase in Vicki's apartment, it's further from the wall in the initial shot than when we see it actually smash in the close up.
Continuity mistake: Towards the end of the gunfire exchange scene at Axis Chemicals - from the point at which Joker was pondering an escape on the catwalk until Batman turns around to see the shadow of a police officer closing in on him - notice how the exit bears an altogether different appearance between views. (00:27:35 - 00:29:40)
Continuity mistake: In the scene where Vicki introduces herself to Knox at Gotham Globe, the newspaper she's reading changes between shots. (00:13:25)
Continuity mistake: When the man is knocked out in the alley he changes positions between shots.
Other mistake: A lady is looking / smiling straight at the camera and crew at the start of the film when a character is trying to get a cab.
Continuity mistake: In the scene where Joker smashes a mirror on the surgical instrument table, a large fragment breaks off and hits the floor. That fragment somehow appears on the tabletop seconds later. (00:35:10)
Answer: He was suspicious, but when she first entered the penthouse and Grissom said to leave their business meeting, he noticed the sexy facial exchange Alicia and Jack gave each other.