Character mistake: Officer Rachel Witten tells Eddie the gun she has belonged to her dad when he was on The Job. In episode 15 of season 8 Gormley tells Frank that Witten's dad was a FDNY Battalion Chief and her uncles were police officers. In episode 12 of season 9 Officer Witten's dad Sgt. Scott Witten speaks to Frank about being on The Job. It's unlikely and not normal for someone to become a firefighter after being a police officer or vice versa or do both jobs.
Continuity mistake: Jenko and Witten confront a guy creating a commotion in a market. The felon runs past them out to the sidewalk where, somehow, the two cops are waiting and grab him.
Other mistake: Eddie and her partner respond to a store selling sandwiches with contradicting prices. For example, grilled cheese is listed for $3.99 and $6.25, and tuna melt is $5.50 and $5.99, and BLT is listed twice on the same sign. (00:03:10)
Continuity mistake: During the family meal, Jaime takes a bite of food, then immediately in the next shot, both arms are down, and he is talking and not chewing.
On the Arm - S12-E11
Character mistake: Lt. Gormley mentions a Captain (Captain Terrell) taking items On the Arm that is a 33-year veteran. Captain Terrell tells the PC that when she was 12 her brother was wounded in Iraq. If she has 33 years on The Job, there's no way she could have been 12 during the war in Iraq. The Vietnam War would have been more appropriate for the age and background of Captain Terrell. (00:00:21 - 00:14:33)
Continuity mistake: Jamie runs to a car which has crashed and has a light pole across its roof. The light pole disappears in the closeups as Jamie struggles to free the driver.
Character mistake: After the explosion, Danny calls for a medic. At the hospital Baez ask Danny if the reason he called for a medic instead of a bus was it because it reminded you of something that happened. Danny was a Marine, so if he was having a flashback he would have called for a corpsman instead of a medic. Medics are in the Army, not the Marine Corps. (00:06:50 - 00:54:22)
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.