One night, on learning that some members of his audience were German, he said, “Forty million Jews in this country, and I got four Nazis sitting here in front waiting for the rally to start.” He said that America needed Italians “to keep the cops busy” and blacks “so we can have cotton in the drugstore,” and that “Asians are nice people, but they burn a lot of shirts.” He might ask a man in the audience, “Is that your wife?” and, when the man answered yes, respond: “Oh, well. Keep your chin up.”

As brutal as his remarks could be, they rarely left a mark. (“I’m not really a mean, vicious guy,” he told an interviewer in 2000.) Sidney Poitier was said to have once been offended by Mr. Rickles’s racial jokes. But in “Mr. Warmth: The Don Rickles Project,” a 2007 documentary directed by John Landis, Mr. Poitier sang Mr. Rickles’s praises.

Recalling the first time he saw Mr. Rickles perform, Mr. Poitier said: “He was explosive. He was impactful. He was funny. I mean, outrageously funny.”

Mr. Rickles got his first break, the story goes, when Sinatra and some of his friends came to see him perform in 1957 — in Hollywood, according to most sources, although Mr. Rickles said it was in Miami. “Make yourself at home, Frank,” Mr. Rickles said to Sinatra, whom he had never met. “Hit somebody.” Sinatra laughed so hard, he fell out of his seat.

Mr. Rickles was soon being championed by Sinatra, Dean Martin and the other members of the show business circle known as the Rat Pack. Steady work in Las Vegas followed. But he was hardly an overnight success: He spent a decade in the comedy trenches before he broke through to a national audience.