Character mistake: When the private investigator shows Castle and Beckett the VIP Liaisons website, "Liaison" is misspelled "Liason." (00:17:40)
Continuity mistake: Castle asks his daughter what she would do if she needed a quarter of a million dollar, just like the murdered victim. She answers that she would ask him for it. Watch her necklace. It jumps in and out of her top. (00:31:10)
Continuity mistake: Alexis has just walked into Castle's study and is relaying a message from his publisher to her dad. She holds the note differently in consecutive shots. (00:30:20)
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.