We study a class of Banach spaces which have the property that every continuous convex function on an open convex subset of the dual possessing a weak * continuous subgradient at points of a dense G § subset of its domain, is Frechet differentiate on a dense G$ subset of its domain. A smaller more amenable class consists of Banach spaces where every minimal weak * cusco from a complete metric space into subsets of the second dual which intersect the embedding from a residual subset of the domain is single-valued and norm upper semi-continuous at the points of a residual subset of the domain. It is known that all Banach spaces with the Radon-Nikodym property belong to these classes as do all with equivalent locally uniformly rotund norm. We show that all with an equivalent weakly locally uniformly rotund norm belong to these classes. The condition closest to a characterisation is that the Banach space have its weak topology fragmentable by a metric whose topology on bounded sets is stronger than the weak topology. We show that the space ^oo(Γ), where Γ is uncountable, does not belong to our special classes.

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We provide an explicit recipe for constructing a function on an arbitrary real Banach space whose Clarke and approximate subdifferentials are identically equal to the dual unit ball.Mathematics Subject Classification: Primary 49J52. Secondary 46B99.

Abstract. Under the assumption that there exists in the unit interval [0,1] an uncountable set A with the property that every continuous mapping from a Baire metric space B into A is constant on some non-empty open subset of B, we construct a Banach space X such that (X * , weak * ) belongs to Stegall's class but (X * , weak * ) is not fragmentable.

For a locally Lipschitz function on a separable Banach space the set of points of Gateaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gateaux but not strictly differentiate is of the first category. However, it is well known that the extension of differentiability theory from continuous convex to locally Lipschitz functions is fraught with difficulties. In particular,