Continuity mistake: In season 2 episode 11, Jake is shown doing at least 3 pull ups but in this episode he shows he doesn't know how to do one and Terry has to show him.
Character mistake: Captain Holt calls the Pimento case "Operation 225641441636324" which he said he arrived by placing a numerical value to each letter in Pimento and then squaring it. But the perfect squares couldn't match up to Pimento since there's only 6 perfect squares and Pimento has 7 letters (plus the last 2 perfect squares don't fit the pattern of the previous 4). Nor is "225641441636324" a perfect square.
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.