Deliberate mistake: The newspaper that shows Petrelli's victory has repeated paragraphs through the article. The article seems to be composed of three paragraphs. One paragraph starts with "I was," and two paragraphs start with "The."
How to Stop an Exploding Man - S1-E23
Deliberate mistake: At the plaza, Peter tells Claire to shoot him because there is "no other way". Nathan then comes flying in and declares, "Yes, there is, Claire!" However, Nathan could not have heard Peter and Claire from the distance he flew in and the rushing of air past his ears.
Chosen answer: It's not been made 100% clear. It was originally assumed that he actually consumed part of the brain, but this has turned out not to be the case. Sylar has been shown all along to have an instinctive knowledge of how things work and how to fix them - this is his actual original power and the ability that made him an exceptional watchmaker and repairer. It appears that he uses a similar process when acquiring new powers - he exposes the brain of the target, allowing him to examine it and determine how their power functions. He can then apply that knowledge to his own brain, allowing him to replicate the ability in question.
Tailkinker ★