Geodesic
On the Difference Property of the Class of Pointwise Discontinuous Functions and of Some Related Classes
Canadian journal of mathematics, Tome 36 (1984) no. 4, pp. 756-768

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Let R denote the set of real numbers. For f:R → R and h ∈ R, the difference function Δh f is defined by The function H:R → R is called additive if it satisfies Cauchy's equation Let F be a class of real valued functions defined on R. F is said to have the difference property if, for every function f:R → R satisfying Δh f ∈ F for every h ∈ R, there exists an additive function H such that f — H ∈ FIt was conjectured by P. Erdos that the class of continuous functions has the difference property. This conjecture was proved by N. G. de Bruijn in [1], where the difference property of several other classes was verified as well. (For other references, see [6].) 
Laczkovich, M. On the Difference Property of the Class of Pointwise Discontinuous Functions and of Some Related Classes. Canadian journal of mathematics, Tome 36 (1984) no. 4, pp. 756-768. doi: 10.4153/CJM-1984-043-3
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[1] 1. de Bruijn, N. G., Functions whose differences belong to a given class, Nieuw Archief voor Wiskunde 23 (1951), 194–218. Google Scholar

[2] 2. Erdös, P., Some remarks on set theory. Annals of Math. 44 (1943), 643–646. Google Scholar

[3] 3. Erdös, P., A theorem on the Riemann integral, Indaaationes Mathematicae 14 (1952), 142–144. Google Scholar

[4] 4. Hyers, D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. Google Scholar

[5] 5. Kuratowski, K., Topology (Academic Press, New York and London; PWN. Warsaw, 1966). Google Scholar

[6] 6. Laczkovich, M., Functions with measurable differences. Acta Math. Acad. Sci. Hungar. 35 (1980), 217–235. Google Scholar

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