Plot hole: From a logistic and tactical point of view, the whole police/FBI operation revolving around the diamonds does not make any sense. The agents are supposedly spread around and in great numbers, but everyone is simply chasing him, while Cross is speaking freely through the radio and shares the addresses and instructions so they should be -ahead - of him, they know where he is going! Although, during the action, Cross tends to speaks directly with Jezzie all the time, as if all the other dozens of agents did not exist. Jezzie's presence itself is completely illogical, too, since Soneji ordered that nobody would follow Cross or he will kill the girl, and Jezzie is the one agent that he knows (she's been the kid's bodyguard for 2 years!) and is sure to recognize. She should be nowhere near the chase, but nobody objects at all.
Revealing mistake: Cross is between the first two steps of his tour de force following the instructions for the diamond delivery. McArthur says "Unit six, take "K" Street and double back", and we are shown Jezzie in her vehicle. She moves cutting through a traffic island with a forbidden maneuver, and yet you can see on the asphalt wet marks of 'someone' doing the exact same thing before her recently (obvious residual of the previous take). (01:06:30)
Continuity mistake: Dimitri calls Jezzie from the gates of the embassy, and she leaves Cross and the guy talking in Russian to go to him. During the transition, the soldier moving across the gates walks through the same steps he already walked, and the bodyguard in a trenchcoat is in a different position. (00:59:25)
Continuity mistake: Cross just boarded the train. At the entrance, featured prominently, there's a woman with headphones. To the other side, there's an old pudgy man with a beret, and a black guy in a dark brown suit, looking all serious. Morgan Freeman is between these two guys, but in the POV that follows they are both on the same side, in front of him. (01:09:25)
Other mistake: Megan is kidnapped during the lunch break, and when Cross reaches the Rose residence, his friend from the FBI tells him about the forensics exams not netting any results. When he gets to the crime scene the day after, though, it's swarming with feds who are still doing things like checking the computer stations and even photographing the chalklines of the strangled woman, all sort of investigations that you'd expect to have happened already in the previous 24 hours or so.
Continuity mistake: Cross and Jezzie are en route for Soneji's office but get stopped by a backhanded comment by agent Ben Devine. During the sequence, there are 3 people talking behind the two investigators; they are either on the left or the right side of the corridor, depending on the camera angle. They just want to be in frame. (00:32:45)
Continuity mistake: After the scene with the senator's wife sticking with Cross after the attempt to undermine him, Megan wakes up. Between the actual shot of her and the camera feed the sheets are touching the ground in a different way, and quite evidently the shadows cast in the scene are completely different. (00:24:00)
Continuity mistake: Cross is discussing who is in charge with Kyle, who just handed the evidence to an agent. In the same shot you can see the fed that gets in the background between them. At the cut, he's turned around facing towards the right (of the screen) while he was turned towards the left in the previous shot. And when they walk away, he has now his back turned to them entirely. (00:19:30)
Continuity mistake: Alex Cross encounters his old friend from the FBI, Kyle Craig. He briefly shows him the pieces of evidence; check out of the strap on the recorder; it is dangling in its length at first, and it is up entangled in Cross' fingers in the reverse. And Craig is holding the bagged boot in a different way (the hand is turned the opposite direction) in the two shots. (00:19:10)
Continuity mistake: After Jezzie and the principal have popped by the classroom finding it empty, Soneji is pushing the cart outside. Behind him, a couple of kids walking from left to right are walking in the path two guys walking in opposite direction, and they are ahead. But in the close-up on Michael Wincott, the two have crossed the other pair (those walking in the same direction as Soneji) walking well ahead of the others. (00:10:50)
Factual error: During the movie, Senator Rose's daughter is protected by the US Secret Service. A plot point also sanctions that senator Rose is basically a nobody, he does not hold important roles, nor he was threatened in any way before the fact. His daughter would not be under Secret Service protection then, since it is reserved only to the highest elected officials of the nation, not every single relative of every single House member.
Revealing mistake: During the botched save of agent Fisher, the helicopter with the spotlight is well behind Cross and the others, and yet the light is projected on parts of the bridge who are facing away from that light and in fact blocked by the bridge structure, as if a strong light source was coming from the front of the rescuers. (00:04:40)
Revealing mistake: In the van at the beginning of the movie, Cross is checking out newspaper clippings about the serial killer. When they are shown in close-up for a couple seconds, the two leftmost clippings, that Cross took his time to pick and hang in the van, are obviously the same article with different titles (both in a single column strip, they even have the same layout and division in paragraphs, it's easy to spot even if you don't repeatedly watch or freeze-frame to actually read, and you just need to glance to see that they even have the same author listed). (00:02:50)
Character mistake: Amongst the material collected about Cross in Jezzie's computer, there's a big magazine article related to the case of the previous movie "Kiss the Girls." The article writes in a huge font the title "The CASSANOVA killer." Not a very good magazine, misspelling Casanova like that. Incidentally, her notes also get the word "Temperament" wrong, missing the 'a', and she has a folder called "Dimitiri", getting the name of the Russian kid wrong. (01:28:30)
Character mistake: The computer technician sitting at Megan's desk is "scoping" (her software is in "scoping mode"!) the GIFs the kids have been using to exchange information. Telling Cross what the name means, she says it stands for "Graphics Interchange Formats", while it is Graphics Interchange Format, no S. She also talks about "billions of 0s and 1s" making each image, which would mean billions of bits, which would mean the kids are casually exchanging 100 MB gifs, which are not so likely nowadays, and even less in 2001. (00:31:55)
Continuity mistake: Cross found out that the picture missing is Charles Lindbergh's (like the blackboard says). He says then "Our Mr. Soneji has a sense of humor." The IT agent by him has her arms folded, hands visible at the elbows. Close-up on Morgan Freeman, and her hands are not on display anymore. (00:35:50)
Continuity mistake: Alex Cross is in front of a computer screen, checking out images of the class to spot the missing picture. From behind, Morgan Freeman is in pose with his finger on his temple, then lowers his hand and turns around. In the next shot, he's still holding his hand high as he is turning. (00:34:50)
Character mistake: Cross knows (for some reason) by memory the last words of the Lindbergh kid kidnapper (and killer) and that Soneji/Mercusio referenced in a book he finds in amazingly oblique ways. He calls him "Richard Bruno Hauptmann", while his name was actually Bruno Richard Hauptmann. (00:42:30)
Continuity mistake: Cross begins his running game with the kidnapper. He says "I'm on my way to Sixth and Pennsylvania.", to which Jezzie goes "I'm left and behind ya, Alex." She is from behind using her right hand on the steering wheel, but she has the left on it in the following shot. (01:05:40)