Continuity mistake: When Sullivan and his pal are watching the golden dome building from a bench, from a back shot his friend has his head faced straight, but from the front shot it's sideways.
Continuity mistake: The dead woman on the beach lies with her feet together, but a shot later they are spread apart.
Continuity mistake: During the opening scene at the diner, Costello places a comic book in the grocery bag, and it is well below the top of the bag. In the next shot of the bag, the comic book is much higher in the bag, and in fact is almost rubbing the boy's cheek.
Factual error: In the 'She fell funny' scene, where Costello shoots the young couple, the aircraft that flies by is a 747-400, which first flew in 1988. The scene is set in 1986.
Continuity mistake: When Costello is hitting Costigan's broken wrist, as Costello is leaving the room, the shot shows Costigan crumpling to the floor beside the pool table. He is on the floor as Costello is leaving. But, the next shot shows him back on the pool table crumpling to the floor again from a bird's eye view.
Continuity mistake: When Costello and Sullivan are in the porno theater, Costello's line is "it won't be me who suffers for it."When the tape is played back to Madolyn later, it says "it won't be me who pays for it."
Continuity mistake: When Costello tells the girl he has a hard-on, the position of her fingers on the blue book keep changing.
Continuity mistake: While having dinner, from a side shot we see Sullivan's mouth chuckle. Frame later, from the front, he is serious and smiles half a second afterwards.
Continuity mistake: Just after Billy Costigan is shot, the elevator doors try to close. In the shots where the black guy comes out of the other elevator, the doors aren't moving anymore.
Continuity mistake: When Sullivan is analysing the Providence corpses, the close-up shows both of his hands together, but on the wide shot one is near the corpse, and the other one close to his hip.
Continuity mistake: After Costello smashes Costigan's wrist he hands all of the dollars he has in his right hand. A frame later money reappears in the hand.
Continuity mistake: When Billy is sitting inside Queenan's house, the glass of water is in his hand in one shot, disappears in the next and then returns.
Continuity mistake: The position of Madolyn's spoon while eating the chocolate cake is not continuous in the front and back shots.
Factual error: When Costigan is riding the "T" (Boston subway), he is talking to Queenan on a cell phone. However, they are on the "Red" line, which is obvious because of the red walls in the tunnel. Cell phones don't work on the Red line because it is underground and there are no cell phone routers in those tunnels.
Continuity mistake: When Costigan is with Madolyn in her office. Madolyn doesn't have jacket on but when she runs out after him to give him a prescription she has a jacket.
Continuity mistake: When Sullivan is questioning a bad guy, he sits down with a cell phone right in front of him. But later on, he reaches out to our left to grab it.
Continuity mistake: After Costigan comes out of jail he visits an old lady. From a front shot she is slightly leaning, her left hand unseen. From a back shot the left hand is standing out over his shoulder with a burning cigarette. It should have been visible from the opposite shot at first.
Continuity mistake: When Sullivan crosses the street there's a lot of people going to the right. From the CCTV recording there's just one person.
Continuity mistake: In the final scene of the movie, when Sullivan is walking down the hallway and into his apartment, his belt is light brown. But when he's dead on the floor, it's a dark brown, almost black.
Visible crew/equipment: When Queenan is in the subway and looks to see if he has a tail following him, you can see a boom mic appear and disappear in the top right corner.
Answer: The FBI and police are separate organizations that do not share full information. Only the FBI and Costello himself knew about the informing, and the FBI can only do so much to stop Costello being caught without drawing attention to their relationship. The police and Costello's gang are fighting a separate battle, hence the two moles.