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Boundedness, differentiability and extensions of convex functions

The genesis of the material in this note can be found in the first author’s paper [2], where, among other things, it was shown that weak Hadamard and Frechet differentiability coincide for continuous convex functions on Asplund spaces. This was expanded upon by the first author and M. Fabian in [3] where relationships between various forms of differentiability for convex functions were connected with sequential convergence of the related topologies in the dual space. In late 1993 whilst Simon Fitzpatrick was visiting Simon Fraser University, he played a key role in producing the paper [5] – which among other things connected boundedness properties of convex functions with sequential convergence of related topologies in the dual space. A few years later, S. Simons [28] produced examples of continuous convex functions whose biconjugates are not continuous and asked