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
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\begin{document}$$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$\end{document}Then every continuous convex function g with \documentclass[12pt]{minimal}
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\begin{document}$g \leqslant f$\end{document} on X is generically Fréchet differentiable. If, in addition, \documentclass[12pt]{minimal}
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\begin{document}$\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty$\end{document}, then X is an Asplund space.
