On functions having Cauchy differences in some prescribed sets

On approximately additive functions

“…Let e ∈ (0, 1 2 ), C = [−e, e] and suppose that there is an additive function A : R → R with h(x) − A(x) ∈ Z + C for x ∈ R. Then the function f = h − A satisfies f (x + y) − f (x) − f (y) ∈ Z for x, y ∈ R and f (R) ⊂ Z + C . Thus, by the theorem of van der Corput (see [14, p. 64] or [3,Remark 2] and [5]), there is c ∈ R with f (x) − cx ∈ Z for x ∈ R, which means that h(x) − (cx + A(x)) ∈ Z for x ∈ R. This is a contradiction with the choice of h.…”

“…The subsequent lemma follows from [3,Proposition]; therefore we present it without a proof. It will be useful in the sequel.…”

“…In this way we continue the research that has been started in [3], for h continuous at a point (see also [1,[9][10][11]21]). Here we study the cases where h is Christensen or universally measurable, or with the Baire property, or bounded on some 'large' sets; we also generalize the main result in [3,Theorem] (see Remark 5, Corollary 12 and Proposition 8). Our results correspond to the problem of Hyers-Ulam stability for the Cauchy equation (see e.g.…”

“…So, Corollary 12 generalizes [3,Theorem], where it is assumed that E is a linear topological space, C is compact, K is discrete, and K ∩ (6C) = {0}.…”