Problem
ALG-B3-M05-P018 The Golden Coefficient
#18
★★★★★ Level 5 of 5
Find all continuous \(f:\mathbb R\to\mathbb R\) such that \(f(x+y)=f(x)+f(y)\) and \(f(x)f(y)=f(xy)+xy\) for all \(x,y\).
The first condition gives \(f(x)=cx\).
Continuous additivity gives \(f(x)=cx\). The second condition becomes \(c^2xy=cxy+xy\) for all \(x,y\). Hence \(c^2=c+1\). We get \(c=\frac{1+\sqrt{5}}{2}\) or \(c=\frac{1-\sqrt{5}}{2}\). Both functions work.
A mixed condition fixes an unusual coefficient.