Restricted Weak Upper Semi-continuity of Subdifferentials of Convex Functions on Banach Spaces

Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X ^* there exists δ > 0 such that $$partial f(y)subsetpartial f(x)+V;;{rm for;all};yin X;{rm with};|y-x| Then every continuous convex function g with $g leqslant f$ on X is generically Fréchet differentiable. If, in addition, $limlimits_{|x|rightarrowinfty}f(x)=infty$ , then X is an Asplund space.