Problem
ALG-B3-M09-P013 Order and Product
#13
★★★★☆ Level 4 of 5
Let \(f\) be nondecreasing, additive, and satisfy \(f(xy)=f(x)f(y)\). Find \(f\).
First obtain \(f(x)=cx\).
A nondecreasing additive function is linear: \(f(x)=cx\), \(c\ge0\). Then \(cxy=c^2xy\) for all \(x,y\), so \(c=0\) or \(c=1\). The answers are \(f\equiv0\) and \(f(x)=x\).
Mixed condition with order.