Problem

Source: Romanian National Olympiad 2017, grade x, p.2

Tags: function, algebra



A function $ f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q} $ has the following property: $$ f(xy)=f(x)+f(y),\quad x,y\in\mathbb{Q}_{>0} $$ a) Demonstrate that there are no injective functions with this property. b) Do exist surjective functions having this property?