09/02/2014 AT 02:00 PM EDT
It's been almost a week since Joan Rivers stopped breathing during a surgical procedure, but her daughter, Melissa, continues to be hopeful about her recovery.
The 81-year-old comedian and Fashion Police host remains hospitalized after she stopped breathing during a surgical procedure at a doctor's office in N.Y.C. on Thursday.
Melissa, who flew from L.A. to be by her mother's side, confirms that "at this time she does remain on life support."
"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 Tuesday. " 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."
• Reporting by EMILY STROHM
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