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Let X be a real Banach space with a separable dual and let f be a continuous convex function on X. We sharpen the well-known result that the set of points at which f is not Frechet differentiable is a first category set by showing that it is even a-porous. On the other hand, a simple example shows that this set need not be a null set for any given Radon measure. Let X be an Asplund space. By definition, this means that if f is a continuous convex function on X, then the set of all points at which f is not Frechet differ- entiable is a first category set. The natural question arises (cf. (1)) of finding the most strict sense in which this set is small. In Theorem 1 we give a partial answer to this question in case of a space with separable dual by showing that the set of points of Frechet nondifferentiability of any continuous convex function on such a space is a-porous. It might be worth noting that the proof of Theorem 1 is, as far as we know, the simplest proof of the result of Asplund that every space with separable dual is an Asplund space. For this purpose, one does not even need the notion of a-porosity, since the corresponding part of the proof of Theorem 1 can be read as a proof that each of the sets Am,k is nowhere dense. Let P be a metric space. The open ball with the center x E P and the radius r > 0 is denoted by B(x,r). Let M c P, x E P, R > 0. Then we denote the supremum of the set of all r > 0 for which there exists z E P such that

Fréchet differentiation of convex functions in a Banach space with a separable dual