Problem
ALG-B1-M08-P006 Additivity on integers
#6
★★☆☆☆ Level 2 of 5
Let \(f:\mathbb Z\to\mathbb Z\), \(f(m+n)=f(m)+f(n)\), and \(f(1)=4\). Find \(f(n)\) for all \(n\in\mathbb Z\).
First find \(f(n)\) for positive \(n\), then for negative \(n\).
For \(n>0\), \(f(n)=nf(1)=4n\). Also \(0=f(0)=f(n+(-n))=f(n)+f(-n)\), hence \(f(-n)=-4n\). Therefore \(f(n)=4n\) for all integers \(n\).
A classical model of Cauchy on a discrete domain.