Continuity mistake: When Beckett is interrogating the drug dealer, in shots from behind, only her right hand is on the table. In shots from the front, both of her hands are on the table. (00:09:20)
Continuity mistake: During breakfast at Castle's loft a jug full of orange juice disappears off the kitchen island when Castle answers the phone. (00:01:40)
Nanny McDead - S1-E2
Continuity mistake: At the precinct as they are watching the security footage from the elevator, a small bit of Beckett's hair keeps moving back and forth over her right ear throughout the scene. (00:06:55)
Home Is Where the Heart Stops - S1-E7
Continuity mistake: When Esposito pulls up Evan Mitchell's rap sheet, it says he is 5'9", but in Mitchell's mugshot, the top of his head crosses the 6' mark.
Nanny McDead - S1-E2
Factual error: The establishing shot of the building where the murder takes place features on the foyer quite prominently "715 - The Langham." The story takes place in New York, but The Langham (named after the famous British hotel) is the name of a residential building in 715 S Normandie Avenue in Koreatown, Los Angeles, used in this episode. There is a hotel by that name in New York, not an apartment building, on 5th Avenue. (00:01:00)
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)
Home Is Where the Heart Stops - S1-E7
Factual error: The shots of the West Side towards the end of the episode are reversed: you can see traffic on the West Side Highway moving in the wrong directions. (00:40:10)
A Chill Goes Through Her Veins - S1-E5
Continuity mistake: When Castle and Beckett visit Mr. Davidson, he comes out chasing his two grandchildren - a tall girl in a red jacket and another one who's slightly shorter in a light pink jacket. When the detective and Castle walk up, the grandkids have switched places. When Mr. Davidson says, "Did I scare you," Beckett has her hand in her jacket, but when Beckett says, "Mr. Davidson," her hand is not in her pocket anymore. When Mr. Davidson says, "Why don't we go inside," Beckett is putting her badge in her pocket, but in the shot before she already put her badge in her jacket. Finally, when Mr. Davidson says, "Alright!" the girls have switched places again between shots. (00:09:50)
A Death in the Family - S1-E10
Continuity mistake: Episode 1-10, "A Death in the Family": when Agent Sorensen offers Beckett a part of the sprinkled donut, the location of where he holds it changes between shots. (00:23:50)
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.