Continuity mistake: When Cillian Murphy's character is recording his message prior to entering the 'dead zone', our viewpoint is from within the monitor in front of him. As such, we see the on-screen image he sees, but reversed, floating between our vantage point and his image. At the bottom of the screen are touch-screen function 'buttons', again in reverse, which read (our left to right) 'RECORD', 'SEND', 'DELETE' and 'REVIEW'. But when our view changes to see the screen as we look over his shoulder from behind him, the visible buttons now read (right to left) 'SEND' and 'REVIEW' (he presses 'SEND'). They should be 'SEND' and 'RECORD'.
Factual error: Slingshotting around Mercury as the Icarus does wouldn't work. It may seem counter intuitive, but getting to the sun is actually really hard, as you have to cancel out the earth's orbital speed around the sun (which is about 30 km/s). Now, assuming this is possible, if they would then get caught in Mercury's orbit, they would have to decelerate all over again, only because Mercury is moving much faster than Earth, it's much harder (about 48 km/s). This would make it a needless detour that would only cost massive amounts of fuel, making it entirely pointless.
Factual error: The entire crew gathers to the viewing deck to watch Mercury traverse the Sun across their field of vision, which only takes several seconds. At the speed it's going, it would make a full orbit around the sun in less than an hour. Mercury's actual revolution period is about 88 Earth days, so from that distance you wouldn't see it move at all.