Problem
ALG-B3-M03-P014 Field compatibility
#14
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\) be additive, \(f(xy)=f(x)f(y)\), and \(f(1)=1\). Prove that \(f(x)=x\).
Hint. Prove that \(x>0\Rightarrow f(x)>0\).
If \(x>0\), then \(x=t^2\) for some \(t\ne0\). Thus \(f(x)=f(t)^2\ge0\). Moreover \(f(t)\ne0\), otherwise \(1=f(1)=f(t\cdot1/t)=0\). Hence \(f(x)>0\) for \(x>0\), so \(f\) is increasing. An increasing additive function is linear: \(f(x)=cx\). From \(f(1)=1\), \(c=1\).
Goal: show which conditions actually force a function to be linear or affine.