Problem
ALG-B3-M10-P008 Square of the Function
#8
★★★★★ Level 5 of 5
Let \(f\) be continuous, additive, and satisfy \(f(x^2)=f(x)^2\). Find \(f\).
Linearity first.
Continuous additivity gives \(f(x)=cx\). Then \(cx^2=c^2x^2\), so \(c=0\) or \(c=1\). The answer is \(f\equiv0\), \(f(x)=x\).
Same system over \(\mathbb R\) with regularity.