But more than that, he has made life harder for his party, which after sweeping victories in last year's mid-term elections, has been trying to recast itself as a party of government rather than of perpetual opposition.

"He doesn't love you. And he doesn't love me. He wasn't brought up the way you were brought up, and I was brought up through love of this country," Giuliani said of the president in an address at a private dinner for business leaders last Wednesday.

Since then he has been given the opportunity to walk his statement back, but decided instead to press home the attack.

By Sunday morning his comments were the focus of America's political morning TV programs.

Levelling personal abuse at the current president has a rich history in the Republican Party, and like most traditions it has different branches and codes.

Some assailants found it useful to demonstrate the muscularity of their loathing for the president by going out of their way to belittle him in public.

In 2009 the president addressed both houses of Congress to outline his proposed healthcare changes. The Republican congressman Joe Wilson twice bellowed at the president, "You lie!"

In April 2012 the ageing rock singer and NRA board member Ted Nugent said in a speech at an NRA conference, "If Barack Obama becomes the president in November, again, I will be either be dead or in jail by this time next year."

In response Republican congressman Steve Stockman invited Nugent to be his guest at the next State of the Union address, demonstrating his patriotism by preening alongside a man who had obliquely threatened to murder his president.

Since then Nugent has called Obama a "subhuman mongrel" in an interview.

The Arizona governor Jan Brewer greeted the President as he alighted from Air Force One during a visit to her state in 2012 by waving her finger in his face (before cameras) and chastising him for his immigration and border control policies.

Later she told Fox News she thought he was very thin-skinned.

But by far the richer vein is that mined by Giuliani, in which one dog-whistles the racist right by questioning the president's fitness for office by casting doubt on his nationality, patriotism and religion.

It is an attack that works because of Obama's name, and his skin colour.

This sort of attack began as soon Obama appeared as a viable candidate, but it was codified by the commentator Dinesh D'Souza who sought to legitimise it by dressing it up with half-baked social-psychological theory in his 2010 bestseller, The Roots of Obama's Rage.

In it he argued that Obama hates America, a hatred he inherited from the Kenyan father he famously barely knew.

"This philandering, inebriated African socialist, who raged against the world for denying him the realisation of his anti-colonial ambitions, is now setting the nation's agenda through the reincarnation of his dreams in his son," wrote D'Souza in the article that formed the basis of the book. "The son makes it happen, but he candidly admits he is only living out his father's dream. The invisible father provides the inspiration, and the son dutifully gets the job done. America today is governed by a ghost."

Since the publication of the book D'Souza has given up claims he might have once had to be a serious commentator. He makes his coin by feeding red meat to conservative conspiracy theorists. Last week after Obama appeared on a BuzzFeed video D'Souza tweeted, "YOU CAN TAKE THE BOY OUT OF THE GHETTO...Watch this vulgar man show his stuff, while America cowers in embarrassment."

YOU CAN TAKE THE BOY OUT OF THE GHETTO...Watch this vulgar man show his stuff, while America cowers in embarrassment pic.twitter.com/C9yLG4QoOK — Dinesh D'Souza (@DineshDSouza) February 18, 2015

His language seems carefully loaded, with reference not only to the ghetto, but with the use of the word "boy" rather than "man", to the plantation.

D'Souza's model has since been employed by respectable conservatives who to allude to Obama's "otherness" without getting into the mucky business of race.

Republican figures like Newt Gingrich and Mike Huckabee return the theme time and again.

This is the company Rudy Giuliani has now joined, and he has been welcomed to it with open arms.

Bobby Jindal, the Tea Party governor of Louisiana widely believed to be preparing for a second tilt for the White House called Giuliani to congratulate him and issued a statement of support. Michele Bachmann has offered her support.

Why Giuliani would want to join this club is less clear.

It could be that he sincerely believes all that he said. Many Republicans are appalled by Obama's refusal to explicitly link Islam to terrorism. Many were infuriated by Obama's recent address at the annual National Prayer Breakfast, in which he raised dark moments in Christian history.

A more cynical view is that ugly attacks on Obama have proved a good career move for washed-up politicians. It helps maintain a public profile and can even lead to a paid gig on Fox News.

There is a whole publishing industry based on spurious attacks on the Obamas.

Donald Trump, a boorish real estate developer and TV host, has been able to remain a part of the political discussion in America simply by endlessly questioning Obama's loyalty and place of birth.

The constituency for this sort of attack is large. A 2010 poll by Pew Research found 17 per cent of respondents believe Obama is a Muslim, 49 per cent correctly identified him as a Christian and 31 per cent said they did not know.

Whatever the comments do for Giuliani's public profile, they could cause the Republican Party trouble as the 2016 presidential election cycle begins.

Since he spoke out journalists have begun using the "do you believe Obama is a Christian/patriot" question as a sort of Rorschach test for Republicans. Answer the question in the affirmative – or, as we once might have said, correctly – and you risk antagonising the belligerent right. Get it wrong and you might be perceived as right-wing fringe dweller.

One leading Republican contender, the Wisconsin governor Scott Walker, has already been tripped up this way. Walker was present when Giuliani spoke and asked the following morning if he agreed he said, "The mayor can speak for himself. I'm not going to comment on what the president thinks or not. He can speak for himself as well. I'll tell you, I love America, and I think there are plenty of people – Democrat, Republican, independent, everyone in between – who love this country."

Pressed harder he added, "I'm in New York, I'm used to people saying things that are aggressive out there."

Marco Rubio did slightly better when he faced the same question, noting that Democrats aren't compelled to respond when the Vice President, Joe Biden, says something embarrassing, and adding, "Suffice it to say I believe that the president loves America. His ideas are bad."

That answer might prove a template for other Republicans. But they should be concerned that "do you believe the president loves his country?" has joined "do you believe in evolution?" and "do you believe in climate change?" as dangerous questions for Republicans to answer.