Flowers for Your Grave - S1-E1
Continuity mistake: Castle approaches his mother and daughter at the bar and orders "a hit of the bubbly." He kisses his daughter; there is one open notebook stacked on top of a closed red one. The angle changes and the open notebook shifted off the closed one. (00:04:35)
Flowers for Your Grave - S1-E1
Other mistake: Castle's book "Storm Fall" is a bit of an enigma, since it is amongst all the other books in the collection of the scapegoat killer when Kate opens it, but Castle gifts "an advance copy" to her later, while in the very first scene of the show there is a party with the book already in the hands of the public. Reviews and a proper release appear only in the fourth episode.
Flowers for Your Grave - S1-E1
Character mistake: At the end of the episode, Castle finds his inspiration again and there's a word processor page fading in/out with text being written. Funnily enough, whoever typed in overlooked something, since there's a sentence that says "Begfore she felt safe." (00:41:30)
Answer: The networks are now having winter breaks. A mid season end of new episodes, when they come back after the new year, it picks up as if several months have gone by. Beckett suggested Valentine's Day as a "chick flick" to watch as she is moved in. Also some episodes are not scheduled to air in order.