Problem
ALG-B3-M03-P013 Additivity and multiplication
#13
★★★★★ Level 5 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)\) and \(f(xy)=x f(y)+y f(x)\). Find \(f\).
Hint. First use additivity on \(\mathbb Q\).
Additivity gives \(f(q)=cq\). Substitute into the second condition: \(cxy=x\cdot cy+y\cdot cx=2cxy\). With \(x=y=1\), \(c=2c\), so \(c=0\). Answer: \(f\equiv0\).
Goal: show which conditions actually force a function to be linear or affine.