Problem
ALG-B3-M09-P010 Two-Sided Estimate on a Ray
#10
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Let \(f\) be additive and \(0\le f(x)\le x\) for all \(x\ge0\). Prove that \(f(x)=cx\), where \(0\le c\le1\).
From \(f(x)\ge0\) for \(x>0\), get monotonicity.
Nonnegativity on the positive half-line gives nondecreasingness. Hence the additive function is linear: \(f(x)=cx\). From \(0\le f(1)\le1\), we get \(0\le c\le1\).
Controls the coefficient through a ray.