Additivity, subadditivity and linearity: automatic continuity and quantifier weakening

We study the interplay between additivity (as in the Cauchy functional equation), subadditivity and linearity. We obtain automatic continuity results in which additive or subadditive functions, under minimal regularity conditions,are continuous and so linear. We apply our results in the context of quantifer weakening in the theory of regular variation, completing our programme of reducing the number of hard proofs there to zero.