Revealing mistake: During the chase in the mountains, the snow thrown up by bullet impacts is black rather than white, revealing their explosive origins.
Continuity mistake: When Bond gets his X-ray glasses for the first time, he puts them on and looks down at R. In the next shot, his head is up again.
Revealing mistake: Right at the beginning of the boat chase when Bond drives the boat through the waterway in MI6, before he jumps the boat into the river, there is a shot roughly from Q's POV, the boat clearly slows right down before it reaches the end. It couldn't fly as far through the air as shown.
Revealing mistake: Near the end of the movie where Renard and Bond are going at it, Bond is choking Renard and vice versa. You can clearly see Bond's thumbs aren't even touching Renard's neck.
Continuity mistake: In the warehouse sequence Bond is taking cover from gun fire and yells to Zukovsky and Christmas "Get Out!" while holding his Walther P99. A helicopter cuts though the building his in and he drops down to protect himself (he's in the same place) and he's holding another gun (possibly an UZI) Then a couple of scenes later 007 drops down on two bad guys and shoots them with his P99.
Continuity mistake: When they're loading the submarine, and Renard is being kissed, from behind his lips are pushed forward, but when we see his face he's trying to pull back.
Revealing mistake: During the action sequence on the labyrinthine docks on the Caspian Sea, one of the choppers with the rotating blades saws Bond's BMW in half. A cutaway shot reveals Bond nearby, murmuring, "Q's not gonna like this." The next shot is in back of Bond, seeing the newly split BMW fall apart, but Pierce Brosnan has been replaced with a dummy. This is most obvious during the over-the shoulder shot of the BMW falling apart, where Bond's gun is now held in front of him. Also, when the car splits in two, the dummy shakes a bit.
Suggested correction: The over-shoulder shot is not clear enough to reveal a dummy; when the car splits in two, the whole boardwalk around it shakes, as would the real actor/double.
Continuity mistake: When Bond's in the chair having his neck broken, between two shots the handles change position. Also, at one point we hear that the woman turns the handles a bit, but in the next shot they haven't moved.
