Democrat Hillary Clinton sought on Tuesday to keep Republican rival Donald Trump on the defensive a day after their first U.S. presidential debate with accusations he is a sexist, racist and tax dodger, while Trump suggested he would "hit her harder" next time by bringing up her husband's infidelity.

While the New York real estate mogul found himself in another controversy over his fresh insults about the weight of a former beauty pageant winner, Clinton tried to keep up the momentum after her forceful debate performance.

She told reporters that during the debate, Trump "was making charges and claims that were demonstrably untrue, offering opinions that I think a lot of people would find offensive and off-putting."

For his part, Trump, speaking at a rally in the battleground state of Florida on Tuesday, said of the debate: "On issue after issue, Secretary Clinton defended the terrible status quo – while I laid out our plan to bring jobs, security and prosperity back to the American people."

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Monday night's face-off between Clinton, with decades of experience in public life, and Trump, famous as a television personality but running for office for the first time, attracted a record audience for a U.S. presidential debate.

Nielsen data showed 84 million people watched the debate on U.S. television, topping the 80.6 million viewers for the Jimmy Carter-Ronald Reagan TV presidential debate in 1980.

Trump praised himself for not attacking Clinton during the debate about the marital infidelity of her husband, former President Bill Clinton, but said he may take up the attack line going forward. There are two more debates scheduled, on Oct. 9 in St. Louis and Oct. 19 in Las Vegas, ahead of the Nov. 8 election.

"I may hit her harder in certain ways. I really eased up because I didn't want to hurt anybody's feelings," Trump said in an interview on the Fox News program "Fox & Friends."

He added that when Clinton criticized him for his treatment of women, he held back, saying: "I was going to hit her with her husband's women. And I decided I shouldn't do it because her daughter was in the room."

Clinton brushed off Trump's vow, saying: "He can run his campaign however he chooses." She added that "the real point is about temperament and fitness and qualification to hold the most important, hardest job in the world."

Trump himself was still married to his first wife, Ivana Trump, when he had a high-profile affair with Marla Maples, who became his second wife. He eventually divorced Maples and married his third and current wife, Melania Trump.