Let \(f_ 1(n)\) and \(f_ 2(n)\) be additive arithmetical functions. Assume that both \(f_ 1(n)\) and \(f_ 2(n)\) have limiting distributions, one of which is continuous. It is proved that \(h(n)=H(f_ 1(n), f_ 2(n))\) has a continuous limiting distribution whenever H(u,v) is continuous and strictly increasing in both u and v, and satisfies the following condition. For given real numbers a and b, let \(H_{a,b}(u,v)=H(u+a,v+b)\). For a given x, let \(\gamma_{a,b}(u;x)\) be defined by the equation \(x=H_{a,b}(u,\gamma_{a,b}(u;x))\), where the domain of \(\gamma_{a,b}(u;x)\) is the largest possible u-set. We then require that the number of intersections of \(\gamma_{a,b}(u;x)\) and \(\gamma_{c,d}(u;x)\) be finite for all (a,b)\(\neq (c,d)\). The fact that some nontrivial restriction is needed on H(u,v) is shown by the example: \(H(u,v)=u+v\) and \(f_ 1(n)=-f_ 2(n)\), in which case h(n) has a discontinuous limiting distribution. The result generalizes a recent one of \textit{M. B. Fein} and \textit{H. N. Shapiro} [Commun. Pure Appl. Math. 40, No.6, 779-801 (1987; Zbl 0655.10052)], who established the above result for the special case \(H(u,v)=u+e^ v\).