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The Cauchy functional equation is not only the most important singlefunctional equation, it is also central to regular variation. ClassicalKaramata regular variation involves a functional equation and inequality due toGoldie; we study this, and its counterpart in Beurling regular variation,together with the related Go{\l}\k{a}b-Schinzel equation.

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