Problem
ALG-B3-M10-P018 Jensen Plus Involution
#18
★★★★★ Level 5 of 5
Find all continuous \(f\) such that \(f(x+y)+f(x-y)=2f(x)\) and \(f(f(x))=x\).
The first condition makes \(f\) affine.
The first condition gives \(f(x)=ax+b\). Then \(f(f(x))=a^2x+b(a+1)\). We get \(a^2=1\), \(b(a+1)=0\). The answer is \(f(x)=x\) and \(f(x)=-x+b\), where \(b\in\mathbb R\).
System: Jensen plus iteration.