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Emmanuel Macron has accused protesters of exploiting the death of a teenager shot by police at point-blank range.
At a crisis meeting, France's president said more officers would be deployed to contain the violence, but stopped short of declaring a state of emergency.
He urged parents to keep rioting children at home and social media platforms to remove certain content.
France has been rocked by three nights of unrest after Nahel M, 17, was killed as he drove away from a traffic stop.
On Thursday night alone, 875 arrests were made, according to officials. The night before, hundreds more were detained and injured - including 249 police officers.
Mr Macron said that about a third of those arrested for rioting were "young, or very young". Imploring parents to take action, he said it was their "responsibility" to keep any child intending to protest "at home".
"It's not the state's job to act in their place," he told reporters after chairing a security meeting of French ministers.
Mr Macron condemned the violence of the last three days "with the greatest firmness" and said Nahel's death had been used to justify acts of violence - calling it an "unacceptable exploitation of the adolescent's death".
He also called on social media platforms - citing TikTok and Snapchat in particular - to take down "the most sensitive types of content" being posted, and asked company bosses to disclose to French authorities the identities of "those who use these social networks to call for disorder and promote violence".
From Lille and Roubaix in the north to Marseille in the south, shops were ransacked across France overnight, streets were badly damaged and cars set on fire.
Public transport was closed early in some places and curfews enforced, with French Interior Minister Gérald Darmanin on Friday advising that buses and trams should be halted nationwide from 21:00 local time (19:00 GMT).
France's capital has been at the heart of the unrest because Nahel lived in Nanterre, a north-west Parisian suburb, and was killed there just after 09:00 on Tuesday.
He was shot after refusing to stop for a traffic check and died after emergency services attended the scene. A video, shared online in the hours following Nahel's death, showed two police officers trying to stop the vehicle and one pointing his weapon at the driver.
The officer who fired the fatal shot has since been charged with voluntary homicide and apologised to the family. His lawyer said he is devastated.
Nahel's death has reignited debate around the state of French policing, including a controversial 2017 firearms law which allows officers to shoot when a driver ignores an order to stop.
More widely, it has led to questions of racism in the force. The UN's human rights office said the unrest was a chance for France "to address deep issues of racism in law enforcement".
A UN spokeswoman in Geneva pointed to a recent report by the UN committee on the elimination of racial discrimination, which last December expressed deep concern at aspects of French policing, including what the report suggested was the disproportionate use of identity checks and imposition of fines on specific ethnic groups.
Nahel's mother, Mounia, made her own accusations, saying the officer who shot her son "didn't have to kill" him.
"He saw the face of an Arab, of a little kid, he wanted to take his life," she told broadcaster France 5. Nahel was of Algerian descent.
On Thursday, Mounia led a largely peaceful march of more than 6,000 people in Nanterre. Wearing a white T-shirt emblazoned with the slogan "Justice pour Nahel" ("Justice for Nahel"), she said she hoped the tribute would be an opportunity for the community in and around Paris to remember her only child.
By late afternoon, the march had descended into violence, sparking the third night of violence. Police fired tear gas at masked protesters who set fire to various objects, with people thought to have been out on the streets until the early hours of Friday morning.