Problem
ALG-B3-M03-P019 Additive involution
#19
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\) be additive, \(f(f(x))=x\), and \(f\) nondecreasing. Find \(f\).
Hint. Monotonicity turns additivity into linearity.
By monotonicity, the additive function is linear: \(f(x)=cx\). Then \(f(f(x))=c^2x=x\), so \(c^2=1\). Since \(f\) is nondecreasing, \(c\ge0\), hence \(c=1\). Answer: \(f(x)=x\).
Goal: show which conditions actually force a function to be linear or affine.