On additive arithmetical functions with values in topological groups. III

We prove that if $G$ is an additively written Abelian topological group with the translation invariant metric $\rho$ and $$ {1\over{\log x}}\sum_{n\le x}{\rho(\varphi(n),\varphi(n+1))\over n}\to 0\quad(x\to\infty), $$ where $\varphi:\mathbb{N}\to G$ is a completely additive function, then the extension $\varphi:\mathbb{R}_x\to G$ is a continuous homomorphism, where $\mathbb{R}_x$ is the multiplicative group of positive real numbers.