Let \(f_x\) be a sequence of integer-valued additive functions. The author finds a necessary and sufficient condition for the existence of a constant \(C\) such that \[ | \{ n \leq x, | f_x(n)| >C \}| = o(x). \] The main tool is an inequality of \textit{G. Halász} [Acta Arith. 27, 143--152 (1975; Zbl 0256.10028)] for \( | \{ n \leq x, f_x(n)=i \}| \), which implies that \(\sum _{p\leq x, f_x(p)\neq 0} 1/p\) is bounded. ``Probably a similar result holds for sets of real-valued additive functions'', writes the author. Such a generalization would most likely yield the reviewer's condition for convergence to a degenerate law [Stud. Sci. Math. Hung. 14, 247--253 (1979; Zbl 0486.10044)]. The present proof makes substantial use of the assumption that the functions are integer-valued.