Automatic continuity: subadditivity, convexity, uniformity

Summary.

We examine various related instances of automatic properties of functions – that is, cases where a weaker property necessarily implies a stronger one under suitable side-conditions, e.g. connecting geometric and combinatorial features of their domains. The side-conditions offer a common approach to (mid-point) convex, subadditive and regularly varying functions (the latter by way of the uniform convergence theorem). We consider generic properties of the domain sets in the side-conditions – properties that hold typically, or off a small exceptional set. The genericity aspects develop earlier work of Kestelman [Kes] and of Borwein and Ditor [BoDi]. The paper includes proofs of three new analytic automaticity theorems announced in [BOst7].