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

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 Fréchet differentiable is a first category set by showing that it is even $\sigma$-porous. On the other hand, a simple example shows that this set need not be a null set for any given Radon measure.