Continuity: At the end, the crime is reconstructed and in flashback, we see and hear the murderer makes his fake 911 call. But his words do not match the 911 tape played earlier in the episode.
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Trivia
Marcy, the obsessed fan of Brad's crime show, complains bitterly about his series changing its theme song. This was written as a little joke on Monk's real life fans, many of whom protested the show's 2nd season musical change from Jeff Beal's instrumental to Randy Newman's vocal theme. Newman's vocals opened this episode as usual, but after Marcy repeats her "don't change the theme song" mantra at the end, Beal's theme was brought back to play over the closing credits. It's been used at the end a number of times since. See more...
Updates: 
Series 2
Mr. Monk Is At Your Service (season 5, episode 12)
Continuity: Monk finds the magazine from which the butler cut out words for his blackmail note, and matches a scrap with the word "Police" on it to the article it was cut from - only the left and bottom edges of the scrap don't conform at all to the hole cut in the page. They haven't been trimmed more. In fact, the whole piece is much too large to fit into the space.
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