Problem
ALG-B3-M01-P017 Additivity and boundedness
#17
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\) be additive and bounded on \([0,1]\). Prove that there exists \(c\in\mathbb R\) such that \(f(x)=cx\) for all \(x\in\mathbb R\).
Hint. Boundedness gives continuity at zero.
Let \(|f(t)|\le M\) on \([0,1]\). For small \(h\), choose \(n\) so that \(|nh|\le1\). Then \(|f(h)|=|f(nh)|/n\le M/n\), so \(f(h)\to0\) as \(h\to0\). An additive function continuous at zero is linear: \(f(x)=xf(1)\) first for rationals, then by continuity for all reals.
Goal: separate guessing the answer from a complete proof and domain check.