Darboux and shift-compactness paradigms: automatic continuity in the Gołąb-Schinzel and the Goldie equations

Using shift-compactness, the continuity theorems of Baire and Luzin and a variant of Darboux’s boundedness theorem, we deduce directly the continuity of positive solutions of the Gołąb-Schinzel equation and of the kernels of the closely related Goldie and generalized Goldie equation, from appropriate assumptions of the Baire property or (Lebesgue) measurability.