Continuity mistake: Alexis' necklace changes in almost every single shot that it's shown. Sometimes it's a single loop; sometimes it's two; sometimes it's three; sometimes the shade of red is darker or lighter; sometimes the loops are big or small and so on. It's very noticeable.
Continuity mistake: When Castle and the flight attendants are attempting to obtain fingerprints from a suspect, the liquid in the cup that he throws away changes colors. (00:16:25)
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.