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 examine 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 and of Borwein and Ditor. The paper includes proofs of three new analytic automaticity theorems announced in Research Report LSE-CDAM-2007-24
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