Banach Spaces with the PC Property

A Banach space $X$ possesses the PC (point of continuity) property if for any $w$-closed bounded subset $A\subset X$ the identity map $(A,w)\to(A,\|\cdot\|)$ has a point of continuity ($w$ is the weak topology in $X$). We deduce some criteria for Banach spaces to have the PC property and describe (for dual Banach spaces) relationships between spaces possessing the PC property and spaces possessing the RN or the WRN property.