A source tells PEOPLE that the former mayor's condition is still very "touch and go" Rudy Giuliani on July 31, 2025.Credit: SOPA Images/LightRocket via GettyNEED TO KNOWRudy Giuliani was hospitalized on Sunday, May 3, for pneumoniaHis attorney said on Friday, May 8, that Giuliani is improving daily and "making great progress"A source tells PEOPLE that despite him leaving the ICU, the former mayor's health is still very "touch and go"Rudy Giulian…