Schumacher has been in a coma since the accident on December 29 and a family statement released today said he was "still in a waking up process" which could take a long time.
The statement came a day after Germany's biggest-selling newspaper, Bild, reported that Schumacher had contracted a lung infection, and days after rumours that he had died swept social media.
The statement said, "Michael's family would like to again express their sincere thanks for the continuous sympathy coming from all over the world. The good wishes they receive help the family and, we are convinced they also help Michael, who still is in a waking up process.
"The family continues to strongly believe in Michael's recovery and place all their trust in the doctors, nurses and nursing auxiliaries team. The important thing is not the speed of the recovery but that Michael's healing process progresses in a continuous and controlled way."
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Schumacher has been in intensive care in Grenoble University Hospital since being flown there by helicopter following his December 29 accident in the Alpine resort of Meribel.
The German fell and hit his head against a rock after venturing on to a small off-piste section while skiing with his son and a group of friends.
The impact of the collision was sufficiently powerful to split the helmet he was wearing.
Further surgery was carried out on December 30 to reduce bleeding.