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 quantifier weakening in the theory of regular variation completing our
programme of reducing the number of hard proofs there to zero.Comment: Companion paper to: Cauchy's functional equation and extensions:
Goldie's equation and inequality, the Go{\l}\k{a}b-Schinzel equation and
Beurling's equation Updated to refer to other developments and their
publication detail
