MathNet 2009-part-04 — 175 problems (EN–VI) Question 4

Find all functions $g: \mathbb{R} \to \mathbb{R}$, for which there exists a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$, such that $$ f(x + y) = f(x)g(y) + f(y) $$ for all real $x$ and $y$.