  Category-measure duality concerns applications of Baire-category methods that
have measure-theoretic analogues. The set-theoretic axiom needed in connection
with the Baire category theorem is the Axiom of Dependent Choice DC rather than
the Axiom of Choice AC. Berz used the Hahn-Banach Theorem over $\mathbb{Q}$ to
prove that the graph of a measurable sublinear function that is
$\mathbb{Q}_{+}$-homogeneous consists of two half-lines through the origin. We
give a category form of the Berz theorem. Our proof is simpler than that of the
classical measure-theoretic Berz theorem, our result contains Berz's theorem
rather than simply being an analogue of it, and we use only DC rather than AC.
Furthermore, the category form easily generalizes: the graph of a Baire
sublinear function defined on a Banach space is a cone. The results are seen to
be of automatic-continuity type. We use Christensen Haar null sets to extend
the category approach beyond the locally compact setting where Haar measure
exists. We extend Berz's result from Euclidean to Banach spaces, and beyond.
Passing from sublinearity to convexity, we extend the Bernstein-Doetsch theorem
and related continuity results, allowing our conditions to be `local' --
holding off some exceptional set.
