Continuity mistake: When Jake handcuffs himself to Captain Holt, the cuffs around each of their wrists consists of double metal bars connected to a single bar. The direction of the handcuffs changes a few times, going round their wrists the opposite way, which is quite impossible. Then the same thing occurs when Boyle handcuffs himself to Holt. (00:09:35)
Operation: Broken Feather - S1-E15
Continuity mistake: When Gina makes fun of Boyle for wearing a sweater there is a woman behind her talking to cops, when they zoom out it's a man and it changes back when they zoom back in. (00:12:10)
Sal's Pizza - S1-E9
Continuity mistake: When Jake and Charles are discussing why Sal's Pizza is only seventh in mouth feel, Jake's badge is flipped backwards. When Jake mentions Fire Marshal Boone, it's flipped frontwards again. (00:04:41)
Answer: In reality, the Commissioner is a civilian administrator, appointed by the mayor of New York City. He (as of this writing, the role has always been filled by a man; the show gets that part right) is usually someone who has risen through the ranks of the NYPD, but upon becoming Commissioner, is no longer a sworn member of the force (the highest sworn rank is that of Chief). Since Commissioners are appointed, technically anyone can serve in the position, including Deputy Commissioners (who are appointed, in turn, by the Commissioner himself).
Update: On 1 January 2022, Keechant Sewell became the first ever female commissioner of the NYPD.