SCHWEITZER 1977 Problem 7

Let $G$ be a locally compact solvable group, let $c_1,\ldots, c_n$ be complex numbers, and assume that the complex-valued functions $f$ and $g$ on $G$ satisfy $$\sum_{k=1}^n c_k f(xy^k)=f(x)g(y) \;\textrm{for all} x,y \in G \ \ .$$ Prove that if $f$ is a bounded function and $$\inf_{x \in G} \textrm{Re} f(x) \chi(x) >0$$ for some continuous (complex) character $\chi$ of $G$ , then $g$ is continuous.

L. Szekelyhidi