Problem
ALG-B3-M05-P015 Midpoints and an Involution
#15
★★★★★ Level 5 of 5
Find all continuous \(f:\mathbb R\to\mathbb R\) if \(f(x+y)+f(x-y)=2f(x)\) and \(f(f(x))=x\) for all \(x\).
The first condition makes the function affine.
From the midpoint equation and continuity, \(f(x)=ax+b\). Then \(f(f(x))=a(ax+b)+b=a^2x+b(a+1)\). This must equal \(x\) for all \(x\). Hence \(a^2=1\) and \(b(a+1)=0\). If \(a=1\), then \(b=0\), giving \(f(x)=x\). If \(a=-1\), then \(b\) is arbitrary, giving \(f(x)=-x+b\). All these functions work.
Combines Jensen type with composition.