  We show that an arithmetic function which satisfies some weak
multiplicativity properties and in addition has a non-decreasing or
$\log$-uniformly continuous normal order is close to a function of the form
$n\mapsto n^c$. As an application we show that a finitely generated, residually
finite, infinite group, whose normal growth has a non-decreasing or a
$\log$-uniformly continuous normal order is isomorphic to $(\mathbb{Z}, +)$.
