Problem
ALG-B3-M02-P015 Boundedness after substitution
#15
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\) be additive and \(f(x)\ge -1\) for all \(x\in[0,1]\). Prove that \(f(x)=cx\) for some \(c\in\mathbb R\).
Hint. Use the lower bound to obtain an upper bound on a small interval.
For \(t\in[0,1]\), \(f(t)\ge-1\), and \(f(t)=f(1)-f(1-t)\le f(1)+1\). Hence \(f\) is bounded on \([0,1]\). An additive function bounded on an interval is continuous at zero, and therefore linear: \(f(x)=xf(1)\).
Goal: choose the right first substitution and then verify the found function.