Cauchy's functional equation and extensions: Goldie's equation and inequality, the Go{\l}\k{a}b-Schinzel equation and Beurling's equation

The Cauchy functional equation is not only the most important single functional equation, it is also central to regular variation. Classical Karamata regular variation involves a functional equation and inequality due to Goldie; we study this, and its counterpart in Beurling regular variation, together with the related Go{\l}\k{a}b-Schinzel equation.Comment: Companion paper to: Additivity, subadditivity and linearity: automatic continuity and quantifier weakenin

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 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

Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation

The class of 'self-neglecting' functions at the heart of Beurling slow variation is expanded by permitting a positive asymptotic limit function λ(t), in place of the usual limit 1, necessarily satisfying the following 'self-neglect' condition:(Formula presented.)known as the Goła{ogonek}b-Schinzel functional equation, a relative of the Cauchy equation (which is itself also central to Karamata regular variation). This equation, due independently to Aczél and Goła{ogonek}b, occurring in the study of one-parameter subgroups, is here accessory to the λ -Uniform Convergence Theorem (λ-UCT) for the recent, flow-motivated, 'Beurling regular variation'. Positive solutions, when continuous, are known to be λ(t) = 1 + at (below a new, 'flow', proof is given); a = 0 recovers the usual limit 1 for self-neglecting functions. The λ-UCT allows the inclusion of Karamata multiplicative regular variation in the Beurling theory of regular variation, with λ (t) = 1 + t being the relevant case here, and generalizes Bloom's theorem concerning self-neglecting functions