How much has South Africa changed? And what does the future hold?

The lights are on but despair still grips Mandela's Dark City. David Blair asks the people of Alexandra, the township near Johannesburg that was home the young Nelson Mandela.

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Electricity has arrived in the corner of Alexandra township where Nelson Mandela came to live in 1941 - but that is the one and only physical improvement of the past 72 years. The tiny red-brick house, which served as Mandela's first home in Johannesburg, stands locked and empty, but otherwise unchanged since he rented its single room as a poor 23-year-old.

All around the former president's old front door live ordinary South Africans who would have been his neighbours in that era. "Alexandra occupies a treasured place in my heart," he wrote in his memoirs. "Its atmosphere was alive, its spirit adventurous, its people resourceful." Mandela now lies in intensive care in a hospital 30 miles away, from where his grandson said yesterday that there was "hope that he is going to recover soon".

He rose from Alexandra to the presidency of South Africa, but the resourceful people he left behind still endure much the same privation and squalor. They cheered Mandela's release from prison in 1990, voted for him in the first free election four years later and celebrated his accession to power.

Today, they will pray for his recovery as he endures his eighth day in hospital as a stricken 94-year-old. Yet as Mandela's life reaches the final pages of its last chapter, they also point out that Alexandra is just as troubled and as poverty-stricken as when he lived here. At that time, Alexandra had no electricity and its squalid streets were popularly known as Dark City.

Today, the lights are on - but that change began under the apartheid regime in the 1980s. Since Mandela led the African National Congress (ANC) to power in 1994, there has been no improvement whatever in physical living conditions in the area near his old home.

Thanks largely to him, Alexandra's people now have the right to vote and to say whatever they wish. They can read uncensored newspapers and campaign for the opposition if they so choose. They do not live in fear of arbitrary arrest by brutal security forces, as they did under apartheid when their township - a rebellious place squashed dangerously close to the white suburbs of Sandton and Bramley - was singled out for persecution.

Because of the former occupant of a house on 7th Avenue, they possess the full range of political freedoms. But the inconvenient truth is that their lives remain full of grinding hardship. Hazel Motlana, 51, lives a few steps from Mandela's old front door.

What she calls her "shack" is not made of the sturdy red brick of the former president's old home, but consists of corrugated tin sheets, fashioned into four walls and a roof. She washes clothes using an outdoor communal tap, just as she would have done in Mandela's day. "We hoped for a better life and, no, we didn't get it," said Motlana.

"Last year, they promised to give us houses. But we are still waiting. Nothing has changed. We are still waiting for the houses, for the jobs. We are so disappointed. We blame our government - this ANC government." Motlana still counts herself an ANC supporter - indeed she wears the party's yellow T-shirt - and she does not blame Mandela personally. His 27 years in jail and the final victory over apartheid earned him a permanent place in her heart. Like many South Africans, she calls him by his clan name, Madiba.

And she votes for the ANC because it remains, despite everything, the party of liberation. But Mrs Motlana does not disguise her bitter disappointment, particularly with President Jacob Zuma. "We don't blame Madiba," she said. "We were expecting something better from Jacob Zuma."

Inside her home, her family of four children, aged between 21 and 30, and two infant grandchildren, inhabit a space measuring perhaps five paces long and two wide. There is one bed, where Motlana's three daughters and the grandchildren all sleep.

At night, an ancient armchair and an old table are pushed aside to make space for a mattress on the floor, shared by Motlana, her husband, Solly, and their 30-year-old son. Everyone living nearby shares the same long-drop communal lavatories, just as they would have done in Mandela's time.

In 2006, the former president visited his old home and the entire neighbourhood, including Motlana's family, turned out to see him and to applaud. But she has lost hope that the end of apartheid would change her life. "I don't think South Africa will get better," she said simply. Her daughter, Fortune, 24, is more optimistic - and her life shows that many doors have opened in the new South Africa that Mandela brought into being.

She is a student at the University of South Africa, reading development studies. But funding her battle for a degree is a struggle. Fortune Motlana earns pounds 130 per month from a part-time placement at the home affairs department. But that will end next month - and she has no idea how she will replace the lost income.

Corruption now penetrates every arm of the South African state. Without the right connections, getting a permanent job with the government is exceptionally difficult. "For now, we do have opportunities, but for some of us, they are limited," said Motlana. "The first thing is corruption in government - it is degrading our opportunities. I've been applying and applying for jobs so I can fund my studies. But nothing."

Unemployment remains the pre-eminent social problem: official figures show that 25.2% of adult South Africans are out of work. If you include those who have given up looking for a job, that figure rises to 36.7%. And it is worsening, with unemployment rising by 100,000 in the past six months.

When Mandela lived in Alexandra, he earned pounds 2 per month as a clerk at a law firm. The rent swallowed a third of that sum, leaving him with agonising choices. Often, he opted to save the bus fare by walking the 12 miles to and from the office in the city centre. The man who now possesses a vast collection of designer shirts owned one old suit in Alexandra, itself a gift from his employer.

Mandela later wrote that "for some time I lived literally on plain bread and cold water". All these dilemmas remain familiar to Alexandra's 170,000 inhabitants. There have been some improvements. The ANC government has built rows of new houses complete with running water on the eastern fringe of the township. Zuma has promised to give the poor 220,000 new homes across South Africa every year.

Yet the tower blocks of Sandton, housing the Johannesburg Stock Exchange and one of the biggest shopping malls in the southern hemisphere, loom near the shacks of Alexandra, providing a vivid illustration of the yawning divide between rich and poor.

As Mandela's life nears its end, many South Africans question whether he did enough to help the township-dwellers who struggled and voted on his behalf. He spent too much time soothing the fears of the white minority and allowing them to keep their property, and not enough helping the poor, or so goes the refrain.

In 2010, his former wife, Winnie, endorsed this critique, saying: "Mandela let us down. He agreed to a bad deal for the blacks. Economically, we are still on the outside." As he lies in intensive care, the harsh truth is that five million of his compatriots - about 10% of the population - still inhabit shacks. Does Motlana have hopes for the future of her two-year-old grandson, Rosalanabho? "Maybe his father can take him elsewhere, out of this country," she replied. "Because no, South Africa will not change."

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