Let $f(n)$ be an additive number-theoretical function. A sufficient condition is given for complete additivity of $f(n)$: this condition, involving the behaviour of $\delta(n) = f(n+1) - f(n)$, requires $\delta(u) \to 0$ as $u \to +\infty$ throughout an increasing sequence of positive integers $u$, having “upper density” I (i.e. ($\limsup \left( \sum \, I : u \le x \right) / x = I$) and not necessarily “density” I.