Investors cheered the appointment of U.K. Secretary Theresa May as leader of the ruling Conservative party as one of the few pieces of good news since the Brexit vote on June 23. David Cameron chaired his last government meeting as Prime Minister on Tuesday and will offer his resignation to the Queen on Wednesday.

The pound perked up, trading above $1.31 Tuesday morning, 1.15% higher against the dollar on the day, although still around 12% lower compared to its level before the referendum.

However, investors should be aware that the challenges are only starting for Theresa May, considered a safe pair of hands by many analysts and politicians alike. She will need to take her country out of the world's biggest trading bloc after tense and sometimes acrimonious exchanges between her party colleagues and EU leaders. Here are the main stumbling blocks she needs to overcome:

Her own convictions. May has been a supporter of the Remain campaign and this is partly why the markets cheered her appointment. There have been calls in the U.K. to organize a second referendum -- the government rejected a petition signed by four million people requesting it -- or for the government to take the referendum result, which is not binding, as an advisory opinion only. Theresa May has been quick to declare that "Brexit means Brexit."

Her own party. The Brexit issue has divided the Conservative party like nothing else could, and it is not yet clear if everybody has rallied around May. She needs to have a team in place in a few days, and the team needs to be strong enough and experienced enough to support her in negotiations with highly experienced politicians like Germany's Angela Merkel and France's Francois Hollande. It also needs to be made of people who are pragmatic enough and wise enough to put the country's interests higher than their own or the party's -- which is not what has happened so far.

Defining a clear U.K. position. There is no precedent for what the U.K. is trying to do. Yes, Greenland withdrew from the European Economic Community (EEC) in 1985, but it did so as a territory not a state; it did not have the myriad connections that Britain has with the bloc and it still has to abide by many EU rules today. Before that, Algeria withdrew completely in 1962, but it did so as it became independent from France so the process was not that complicated.

The market is hanging its hopes on the new Prime Minister striking the right note. "May could possibly kick triggering Article 50 way into the long grass, or go for the Norwegian model and allow free movement in exchange for access to the single market. This kind of 'Brexit-Lite' might well please the markets -- which had widely priced in and were largely relying upon a Remain victory before the shock result," Nigel Green, founder and CEO of financial consultancy deVere Group, said.

Selling the proposal at home. Once a U.K. position regarding the withdrawal is defined, it will need to be "sold" to the British public. Despite the Conservative party's apparently easy acceptance of her as a leader, it is likely that divisions will appear when it comes to accepting her Brexit proposal. The same goes for the wider public, despite May's reassurances that she wants to unite the country.

One of the problems the U.K. has always had with its membership in the EU was that a lot of British people felt they had been "mis-sold" a political union back in 1975 when they voted to stay in, when in fact all they wanted was just access to the common market. This is despite the fact that at the time, the Treaty of Rome promising "ever closer union" and guaranteeing freedom of movement for EU workers was in force. The same thing risks happening again, as the British public will not get a chance to vote on every single detail of the withdrawal proposal, so accusation of Brexit being "mis-sold" could arise either now or in the future.

Selling the proposal to the EU. Freedom of movement for EU workers will be the single most important issue of the negotiations. This is not because it is the most important for the U.K. economy, but because for years EU workers in the U.K. have been used as scapegoats by a big part of the media and politicians for everything that went wrong with the economy. If housing is expensive, immigration was blamed for raising demand; if hospitals are crowded, immigrants were blamed for "putting pressure on the NHS" despite cuts in the health budget and hospital closures; if there aren't enough teachers, immigrants' children are blamed for "putting pressure" on schools.

Curtailing EU workers' freedom of movement will therefore be a crucial part of May's withdrawal proposal, but it is hard to see how the EU will accept it. Countries that already have deals with the EU, such as Norway or Switzerland, are obliged to agree to EU nationals working in these countries on the same terms as local citizens. It is difficult to believe that the EU will allow the U.K. to restrict its own citizens' rights while maintaining unhindered access to the common market for Britain.

Keeping the economy ticking along. The U.K. economy suffered from big imbalances even before Brexit. It has big current account and budget deficits, a property bubble and a big welfare budget swallowing resources. Chancellor of the Exchequer George Osborne announced he scrapped plans to cut the budget deficit to zero by 2020 following the vote, while Bank of England Governor Mark Carney is expected to announce an interest rate cut at the bank's meeting this coming Thursday.

The central bank already scrapped a requirement for banks to keep more capital, in order to help them lend more. However, investors' confidence has taken a big hit, and any wobble in negotiations is likely to frighten even more investors away. Already, commercial property funds have slammed the gates shut on investors who wanted to withdraw their funds, raising fears of a property market crash. Theresa May will need all the help she can get to restore markets' trust in the U.K.

Editor's Note: This article was originally published at 9 a.m. EDT on Real Money on July 12.