Revealing mistake: In the extended version, when Luthor fires snow at Superman, the icicles bounce, rather than shatter, revealing they are some type of foam or plastic balls.
Continuity mistake: When Krypton is exploding, Superman's mother looses her balance and falls backwards. In a new angle a frame later she is up-straight loosing balance again.
Continuity mistake: The first shot of Lois' red car shows it clean and shiny, a second later it's all dusty and dirty.
Continuity mistake: When the burglar falls from the building, his flashlight is sometimes swinging, but other times it's flat against his body revealing the camera was tilted.
Continuity mistake: When Luthor smashes the green car, its wreckage is different in the shot when the soldiers arrive: The hood is open wider and the windows are broken.
Continuity mistake: During the trial, Zod and Ursa swap places in a couple of brief shots.
Continuity mistake: At the end of the football game, a guy with a cap tells Clark to have the clothes ready. But his lips are not in synch.
Continuity mistake: When Clark Kent falls out of the window of the Daily Planet and turns into Superman you can see that he is still wearing slacks and not blue tights.
Revealing mistake: When Otis first arrives in Luthor's lair beneath the train station, he knocks a table lamp over. When the lamp shade comes off the lamp, the bright, exposed light bulb then dims to a less-distracting level for filming purposes. Otis then replaces the shade upon the lamp, and the bulb then instantly goes back to its original level of brightness, so that it will look appropriately illuminated through the shade. The bulb's brightness was manipulated off-screen to conform with the needs of the bulb's effect on the scene.
Continuity mistake: By the cradle, Jor-El and her wife are standing a couple meters apart. In the next shot they appear closer together.
Continuity mistake: While being mugged in the alley, Clark's gabardine keeps swapping between neatly folded to wrinkled between shots.
Continuity mistake: When Lex Luthor calls Superman and the dogs bark, a group of trash cans disappear from the right of the black and white dog.
Continuity mistake: The close shot of the dogs and the garbage bins shows stuff on top of the bins. In the next angle they're gone.
Continuity mistake: When Lois is hanging from the helicopter, she looks down and sees some cars on the pavement, by the entrance of the Daily Planet. From a ground view the cars are gone.
Continuity mistake: The morning Clark leaves the home farm, the sky keeps changing throughout the sequence: Dawning, clouds, sunny, dawn again, cloudy, back to sunny.
Continuity mistake: When Superman saves Lois from the car crash, the bumper's damage changes between shots: sometimes it's dangling on the edge, others it's attached to the car.
Continuity mistake: When Superman flies towards Lex's lair, he flies next to a window where women gather hurriedly to watch him. A woman in blue runs towards the window while a brunette stands next to her. The shot changes and the brunette is two meters behind, running towards the window again.
Audio problem: When Lex is hiding the kryptonite in the chest, Superman walks towards him and says, "You diseased maniac! Did you really think you could hide it from me by encasing it in lead?" but his lips are not moving.
Revealing mistake: Look carefully at the train when Superman is using himself to hold up the track during the earthquake. It's obvious the wheels of the train are not even touching the track as they pass over him.
Continuity mistake: Superman saves Air Force One lifting the wing with his right hand and then salutes with his right-hand, suddenly holding the wing with his left.
Answer: He wanted to see for himself if the stories were true. Some reporters tend to exaggerate the facts and if he had any other weaknesses. He couldn't be sure the kryptonite would work.
While I don't exactly disagree that Lex wanted to see for himself how invincible Superman is, I don't think that's the main reason why he did it. The bullets and fire were simply a charade to make it look like this was his security system. Keep in mind, he wanted Superman to enter his lair because the real trap was the Kryptonite that he had in the lead box.
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