Problem
ALG-B3-M03-P011 Boundedness on an interval
#11
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\) be additive and \(|f(x)|\le10\) for all \(x\in[0,1]\). Prove that \(f(x)=cx\).
Hint. Prove continuity at zero.
If \(h\) is small, choose \(n\) so that \(0\le nh\le1\). Then \(|f(h)|=|f(nh)|/n\le10/n\), which tends to zero as \(h\to0\). Thus \(f\) is continuous at zero. A continuous additive function is linear: \(f(x)=xf(1)\).
Goal: show which conditions actually force a function to be linear or affine.