Factual error: The team is discussing the location of the Widmark hotel which the detective says is downtown. Beckett then looks at the map and locates it only 5 blocks from E 72nd street, which is UPTOWN in NYC. Furthermore the map shows E 72 intersecting Broadway which is on the west side of Manhattan. This entire mapping exercise is incorrect.
Continuity mistake: When Castle and Beckett are theorizing about Melanie's relationship with Senator Bracken, Beckett's right hand is visible holding her cell phone. When the shot cuts, her hands are suddenly in her coat pockets, with her phone nowhere in sight. (00:06:25)
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.