Problem
ALG-B3-M10-P006 Golden Product
#6
★★★★★ Level 5 of 5
Find all continuous additive \(f:\mathbb R\to\mathbb R\) such that \(f(x)f(y)=f(xy)+xy\).
After linearity get \(c^2=c+1\).
Continuous additivity gives \(f(x)=cx\). Substitution: \(c^2xy=cxy+xy\). Hence \(c^2=c+1\). The answer is \(f(x)=\frac{1+\sqrt{5}}{2}x\) or \(f(x)=\frac{1-\sqrt{5}}{2}x\).
An irrational coefficient is allowed over \(\mathbb R\).