Continuity mistake: Frasier is based on the character originally created on Cheers. In Fraiser, he lives with his father who is a retired police man. On Cheers, Frasier once mentioned his father was a scientist. It's also worth noting that in one episode of Cheers, John Mahoney, who was to later play Frasier's dad, turns up as a pianist. Frasier also said on Cheers that his father was dead. When Ted Danson made a guest appearance on Frasier, Frasier admitted that he had lied about his father since the two had been arguing so much.
Continuity mistake: Across the show, Marty is supposed to be about 21 years older than Frasier (he turns 65 in season 5). There are at least two problems with his age - Frasier is supposed to have been born in March 1952 and in Season 5 (Roz and the Schnoz) Marty mentions going on shore leave as a single guy in 1952. Even if he was confused about the year, Frasier didn't call him on it. He must have done just a short stint in Korea as he was already a cop when he met Frasier's mother - they met at a murder scene, and he must have been 20 years old at the most. She was already a full fledged psychiatrist.