Continuity mistake: The door to Dexter's apartment opens out toward the walkway. However, in all subsequent episodes the door opens into his apartment.
Other mistake: When Dexter is running Matt Chambers' fingerprint though his computer, he brings up four newspaper articles where the names Matt Brewster and Matt Rasmussen are used. In the final article, the headline reads "Matt Rasmussen of South Boston arrested", but the body of the article uses the name Matt Brewster exclusively. Other than the first few words, the text of the final article is identical to that of the previous one. (00:33:50)
Revealing mistake: Sgt. Batista shows Dexter a dismembered leg with the hip bone protruding. The hip bone looks ridiculously fake, with plastic seams and the pin used to attach it to anatomy models visible. (00:26:05)
Answer: No, Harry is not the real father. While he did have an affair with Dexter's biological mother (Laura Moser), this was when she was his C.I. Dexter was already born at that point.