Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). If there is a constant $c$ such that $|f(n+1)-f(n)|<c$ for all $n$ then must there exist some $c^{\prime }$ such that$$f(n)=c^{\prime }\log n+O(1)?$$