Problem
ALG-B3-M03-P010 Monotone additivity
#10
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\) be additive and nondecreasing. Prove that \(f(x)=cx\) for some \(c\).
Hint. First prove the formula on \(\mathbb Q\), then use rational approximations.
On rationals, \(f(q)=qf(1)\). Let rational sequences \(r_n\uparrow x\), \(s_n\downarrow x\). By monotonicity, \(f(r_n)\le f(x)\le f(s_n)\), so \(r_nf(1)\le f(x)\le s_nf(1)\), with the sign handled by two-sided approximation. Passing to the limit gives \(f(x)=xf(1)\).
Goal: show which conditions actually force a function to be linear or affine.