Continuity mistake: When Elliot is unpacking her books at the apartment, she places a brown book on top of the pile next to her. In the next shot, there is a green book on top of the pile.
Revealing mistake: When Elliot and JD are at the baseball nets, Elliot gets hit by a baseball. If you look closely, particularly in slo-mo, you can see that the ball goes behind her head, and is probably digitally removed from coming out from the other side.
Continuity mistake: The second time that Turk and Elliot are studying in the apartment, papers appear on the table in front of Elliot from nowhere.
Continuity mistake: When JD and Cox are out having a drink and JD is about to leave and says "Look, Dr. Cox. Don't get me wrong. I love getting drunk in my jammies", there is no glass anywhere near Cox. When it cuts Cox suddenly raises a glass to his mouth.
Answer: No, it was a fake, they used a stamp.
It is real.