Revealing mistake: When the woman at the bar brings Jack the mustard which she forgot, Jack puts his napkin in front of his mouth. A fraction of a second before that, you can see the down-side of his face which should be invisible. Instead of nothingness you see it's all black. That is material used for the special effects, but they didn't 'make it invisible'. (00:06:50)
Other mistake: At the end when the naked but invisible hero runs from the police but is given away by his footprints in the snow, the footprints are of shoes, not bare feet.