Problem
ALG-B1-M08-P017 Monotone additive function
#17
★★★★☆ Level 4 of 5
Let \(f:\mathbb R\to\mathbb R\), \(f(x+y)=f(x)+f(y)\), and suppose \(f\) is nondecreasing. Prove that there exists \(c\ge0\) such that \(f(x)=cx\) for all \(x\).
First prove the formula for rationals, then squeeze a real number between rationals.
Let \(c=f(1)\). On rational numbers, \(f(q)=cq\). Since \(1>0\), monotonicity gives \(c=f(1)\ge f(0)=0\).
Let \(x\) be real. For any rational \(r
The first serious warning: on \(\mathbb R\), regularity is needed.