Hong Kong -- The leader of Hong Kong has apologized in person for the first time for the chaos and violence of the past week. Massive demonstrations forced her to hold up a controversial bill that would have paved the way for extraditions from the island to the China.

Protesters fear it could allow people accused of crimes in the semi-autonomous Chinese region to be transferred into the mainland's opaque legal system.

CBS News correspondent Ramy Inocencio reports that Hong Kong Chief Executive Carrie Lam, the Beijing-appointed leader of the island, offered her "most sincere apology" and said she was "sad" about the recent mass protests and violence. On Sunday some 2 million people took to the streets in Hong Kong, demanding a complete retraction of the extradition bill.

"I personally have to shoulder much of the responsibility," she read from a prepared statement at Hong Kong's Central Government Offices. "I offer my most sincere apology to all people of Hong Kong."

Inocencio said Lam's apology only brought more questions about her ability to carry on in the role.

The idea of losing face -- being publicly embarrassed -- is a major cultural touchstone in China. The fact that Lam showed her face at all on Tuesday was surprising, said Inocencio, but it appeared to be too little, too late for protesters, who want her to withdrawal the bill and step down.

Asked by CBS News during her news conference whether her decision not to retract the bill might deepen distrust in her among Hong Kong residents, Lam said she would "not proceed again with this legislative exercise if these fears and anxieties cannot be adequately addressed."

After the news conference, protest organizers from the loosely organized Civil Human Rights Front said they did not accept her apology, as Lam hadn't met any of their demands.

Asked by @RamyInocencio if she was a habitual liar, as she would not "retract" the bill, Lam repeated that it will expire at the end of the 2020 legislative term: "I will not proceed... if these fears and anxieties could not be adequately addressed." https://t.co/3bP6ZAM1FB pic.twitter.com/CtaNGXg3rY — Hong Kong Free Press (@HongKongFP) June 18, 2019

"Disappointed and angry," the front promised to continue its protests and said it would announce plans on Wednesday.

As Hong Kong's embattled leader strategized over her next step, Beijing kept mum on the debate. The central Chinese government issued two statements of support for Lam over the weekend.

Lam admitted she has a tough job ahead to re-earn the people's trust in Hong Kong: "I've still got much to learn and do in better balancing diverse interests, in listening more to all walks of life, in taking our society forward."

But the chief executive's hands are tied. Even if she wanted to capitulate and withdrawal the bill, which might regain her some of the trust from her society, the government in Beijing could stop her doing so, or simply remove her.