Problem
ALG-B3-M09-P016 Absolute Value Bound
#16
★★★★★ Level 5 of 5
Let \(f\) be additive and \(|f(x)|\le2|x|\) for all \(x\). Find all such \(f\).
The estimate gives boundedness on \([-1,1]\).
The estimate \(|f(x)|\le2|x|\) gives boundedness on \([-1,1]\), hence \(f(x)=cx\). The condition becomes \(|c||x|\le2|x|\), so \(|c|\le2\). The answer is all \(f(x)=cx\), where \(|c|\le2\).
The estimate both regularizes and bounds the coefficient.