During the Third Act Police Captain Battersea is roped to a chair by David Vincent. They way Battersea is tied up is completely different from one scene to the next.
When the police car is destroyed, the beam coming out of the ray gun is blue, but when the perspective shifts to show the car, the beam is now red.
The supposed small town on the map is actually Indianapolis a huge city.
Claire tells David that some aliens can feel as deeply with their minds as humans do with their hearts. Feeling with our hearts is only a metaphor not a fact. Claire is an advanced being from an advanced civilization and would know this, so she would never make such a ridiculous statement.