Continuity mistake: As O is helping K into the jet pack harness, he asks her if that is right. She says "Unfortunately it is." In that shot, she is looking up at him right into his eyes. It cuts to a side angle and she is suddenly looking at his mouth and chin. (01:19:15)
Continuity mistake: As J is saying no jet packs, he has both arms out with his hands turned palm down. It cuts to a different angle saying there's a reason they don't have those in the future, and you can see his left hand. It is suddenly gesturing towards the machines but palm up. (01:18:40)
Continuity mistake: After being pulled over by the racist cops, J insults them and they pull him out to frisk him. As they have him bent over the car, one of them pulls out his gun and says "what's this, a concealed weapon". In the shot from behind J turns his head to the left to look at the gun. It cuts to a view of J's face as he says "Hey, it's his grandmother's suit." J is instantly looking straight forward. (00:41:05)
Continuity mistake: After the incident at Wu's restaurant, J walks up to a crowd of people and says "Can I have everyone's attention please?" As he does, he holds up his neutralizer. Just before the cut, you see him raising his other hand and points at the device. It cuts to show the crowd, then back to J. Suddenly he is repeating this exact same motion, the footage repeated as he points to the neutralizer. (00:18:20)
Factual error: When Boris arrives at Coney Island, Cha Cha's Bar and Cafe on the right prominently displays a phone number with area code 718. Area code 718 was created in 1984, 15 years after the 1969 setting. In 1969, all of New York City was in the 212 area code. (00:39:30)
Continuity mistake: When J is drinking his chocolate the glass is next to his mouth in the angled shot from in front, but lowered in the angle from behind.
Continuity mistake: In the last scene at the diner, the various camera angles show the background characters. The view to Will Smith's right shows 5-6 empty seats down to a man at the end of the counter. When Smith and Jones leave, the camera pans left down the same seats and there is Griff sitting 2 stools away from Will's spot. He wasn't there 10 seconds earlier and has partially eaten food in front of him, indicating he was there for a while. And you think Will would have noticed him there after all they recently went through.
Continuity mistake: When K and J drive to the bowling alley, the same junction with a shiny blue car and a man wearing an orange shirt is seen three times through J's window.
Continuity mistake: At the bowling alley, after K polishes the alien's head, the way his hair is messed up changes between shots.
Continuity mistake: When K questions J at the headquarters, K grabs his tie. From the immediate angle his hand is lowered.
Visible crew/equipment: After J arrives in 1969 and stands on the sidewalk at the Empire State Building, look at the doors behind him. As they close and open, the lighting rigs are reflected.
Continuity mistake: As J flies backwards after the fish creature sneezes, he breaks through the plate glass window, and has the bottles in his hands. After the cut, he's just breaking the glass, and the bottles are missing from his hands.
Continuity mistake: When J is being eaten by the fish, his trousers are clean. Right after it growls, a close-up shows his trousers covered in mustard, despite him not shooting it until several seconds later. (00:16:25)
Continuity mistake: When J peeks out The Factory window the broken glass around the frame change amount and shape between shots. (01:01:25)
Continuity mistake: When J puts a tray in the fish's mouth, it swaps from perfect to wrinkled between the first two shots in a time-frame. For the rest of the scene, there's no continuity between the shape it has in the wide angles and in the close-ups. (00:16:20)
Continuity mistake: During the scene where Agent J and K are driving around Midtown Manhattan, the wide shot shows them driving past the red TKTS staircase in the middle of Times Square. Then there is a shot of J inside the car a few moments later, and they pass by the red TKTS staircase again. (00:09:50)
Continuity mistake: When the woman is about to be sucked by the void and fiercely grabs Boris' wrist, the wide and close-up angles show her using her left arm, however in several close-ups it's the right arm shown, as noted by position of her thumb, style of the bracelet (from grey rubber to black with diamonds), and a ring appearing/ disappearing. (00:05:10)
Continuity mistake: When they are leaving the first crime scene (the crashed UFO) in their Ford Taurus they are shown talking with headrests in the vehicle. They arrive at the headquarters for Zed's eulogy. After the eulogy they are back in the Ford Taurus and the headrests are gone. (00:07:40 - 00:09:50)
Continuity mistake: At the diner, J says to the waitress K has apple pie. K is then given some apple pie. In one shot, the waitress is in the middle of giving K the plate with pie and the plate has not reached the table yet. But in the immediate shot afterwards, the pie has already reached the table. (01:03:45)
Continuity mistake: When K slaps the Chinese man with the alien crab his hair gets messy. When the shot changes it's back to brushed. (00:12:45)
Answer: Earlier in the movie, the way K was talking about Boris was different than what J had come to expect of his partner. So J wanted to know what happened between Boris and K at Cape Canaveral that changed him. So more of what did Boris do to K rather than just the report of what K did.