Continuity mistake: The precinct number (12th) is not firmly established until Episode 6 of Season 1 when we see that number painted on the elevator doors at 28:28. Prior to that (Episodes 1, 3, 4 and 5), shots of the outside of the building, including squad car markings, indicate it's the 9th Precinct. The first hint that it might be the 12th is in Episode 4 (01:20) when we see 12th Precinct collar brass on the uniformed officer. But we still see the 9th Precinct building and cars used through to Season 3: 1-1 (06:33); 1-1 (21:03); 1-3 (30:42); 1-4 (11:06); 1-5 (20:44); 1-10 (17:30); 2-3 (24:40); 2-8 (03:50); 2-9 (04:23); 2-15 (28:46); 2-18 (39:17); 2-19 (30:55); 2-20 (19:47); 2-21 (07:01); 2-22 (36:41); 2-23 (5:38); 2-23 (15:15); 2-24 (27:290; 3-1 (25:58); 3-7 (08:53). The first time we see a squad car marked "12th PCT" is 2-22 (02:19), and the first time we see a building marked "12" is 3-13 (33:39).
Continuity mistake: Alexis is trying to get her father to tell her how the interview with the police went as Castle playfully pushes her along the corridor. Castle is holding the bottle of whipped cream in his right hand with the bottle positioned above his daughter's elbow. Cut to the immediate next shot and the bottle is now underneath Alexis' elbow.
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.