Continuity mistake: When Beckett, Esposito and Ryan interview the fruit stand vendor, he states that he saw a bronze Mercedes drive by, to which Esposito replies, "That's Castle's car." In the previous scene at the car crash site, Ryan asks Esposito to check the back corner of the car for damage, which is silver or gray, not bronze, in color.
Continuity mistake: After Beckett gets back from her assassination assignment, she washes up in a bathroom and places the gun she was given in her front waistband near her stomach. Then she goes to meet Lazarus (Vulcan Simmons) in the basement, who realizes she's Beckett. When she pulls out the gun, she reaches behind her back to get it.
Continuity mistake: Beckett's sweatshirt changes position on her left shoulder throughout the scene in which she and Castle are sitting in bed discussing Bigfoot. Her sweatshirt is initially covering her shoulder and then keeps going lower and lower off her shoulder every time the camera comes back to her .
Plot hole: The security camera images of Castle kissing Kyra on the roof are completely inexplicable. The angle at which the images are taken are near enough identical to the angles at which the rooftop scene were shot. There were no pillars, poles or high walls for a security camera to be attached to at those particular angles, and certainly nowhere a photographer could hide, so where did the images come from?
Continuity mistake: When Castle gets back from helping Alexis with her friend and thanks Beckett for taking care of Fagan (Alexis' egg), just after her line, "Yeah well, he was easy. He didn't even fuss when I put him to bed", Ryan and Esposito come in and Ryan is holding a missing persons poster of Elizabeth McGinty. As soon as they walk in, Ryan is holding the poster with the front facing the camera. But two shots later he is holding it with the back facing the camera.
Character mistake: In the scene where Castle and Kate are being told that the victim's SIM card is broken and the last two digits of the phone number are missing, Castle says that they have 99 possibilities, which mathematically is wrong. Obviously the writer didn't count the combination 00. The correct number of possibilities is 100.
Continuity mistake: When Beckett goes to the bathroom to read Castle's book and Castle peers over the top of the cubical to tell her what page the sex scene is on, she is holding the book with her fingers between the pages dividing the book into 2 sections. These 2 sections switch to 3 and then back to 2 between shots.