"On behalf of my mother and our family, we are extremely grateful for all the love and support we've received," Melissa said in a statement. "At this time, she does remain on life support.
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Joan Rivers remains on life support, her daughter Melissa Rivers has confirmed
"I know my mother would be overwhelmed by the continued outpouring of kindness, and I want to thank everyone for keeping us in their prayers."
Melissa has been keeping a vigil by her mother's bedside as she continues to receive treatment at New York's Mount Sinai Hospital.
The 81-year-old was admitted last Thursday after she stopped breathing during surgery on her vocal chords at a private Manhattan clinic.
Joan Rivers stopped breathing during surgery on her vocal chords last week
On Sunday, doctors began the process of lifting Joan out of a medically induced coma. Melissa said she remained hopeful of a recovery – there are fears that the star could be left with permanent brain damage or in a wheelchair.
Joan had shown no signs of illness in the days prior to her operation, recording scenes for Fashion Police and performing gigs as part of her current North America tour.