Visible crew/equipment: As Danny gets out of the car, with the bad guy who shot Erin in the courtroom, holding her hostage and coming out to meet him behind the courthouse, as Danny gets out of the car, in the driver's side car door mirror you can see a crew member in a red shirt holding one of those "scene marker" slap boards. (00:32:40)
Continuity mistake: Near The End of the episode, just after Danny's family is looking at the Statue of Liberty, Danny puts his arm around Linda, and then does it a second time.
The City That Never Sleeps - S4-E2
Continuity mistake: Danny and his partner are driving with a passenger in the back seat. You can see both front seat headrests removed while they are talking in the car. Next shot shows the car from outside and you can see the headrests are on the seats.
Continuity mistake: When Frank goes to see Gallo at the hospital after the lawyer's arraignment, Frank drops a deck of cards on the bedside table beside the bottle of whiskey, but the bottle has vanished in the next shot. (00:41:00)
Continuity mistake: During the courtroom hostage scenes, when Raoul Delgado and Erin are on the phone with Danny there are some closeups of the gun Delgado is holding. After Delgado hangs up on Danny, when he slams the gun into Jack's temple and orders Erin to go with him, in these closeups the gun is noticeably different. (00:30:55)
Continuity mistake: After Danny shoots Delgado, he kneels down and holds Erin at her right side with his back facing the car (driver's door is open), but in the next shot Danny is kneeling at Erin's left side with his front now facing the car, as seen in the live feed Frank is watching on the monitor. The live feed is not backwards, note the open driver's door, etc. (00:34:35)
Continuity mistake: Danny drives up to the police station. Erin is waiting for him, leaning on the trunk of a police car. A different camera angle shows her standing up. Next shot she is leaning against the car again, then next shot she is standing.
Continuity mistake: Frank is talking to Garret in his office. A close up shot shows him sitting at his desk with his elbows on it. A wider angle shot shows him sitting up straight. Next shot shows him with his elbows on the desk again.
Other mistake: Danny says "14 years ago today I kissed Linda" but in previous episodes they were celebrating their 16th anniversary.
Continuity mistake: One second Frank Reagan throws his glasses on his desk, the next after a brief cutaway, they are on his face. (00:36:10)
Continuity mistake: Danny and Linda are hugging after Danny tells the boys to go pay your respects to the WWII Memorial. In the next shot they are standing apart as when he told them to go pay their respects to the memorial. (00:54:34)
Other mistake: Throughout the episode, all the interactions with the characters went from day to night except for Commissioner Reagan chaperoning his grandson's class at the museum. It stayed daylight the whole time at the museum. (00:27:14)
Answer: It's a common cop show plot device. If the police on the show always acted in the most logical way (i.e. not alerting a suspect to their presence) the show wouldn't have the opportunity for an exciting foot chase.