Problem
ALG-B1-M08-P004 Values at 0 and 1
#4
★☆☆☆☆ Level 1 of 5
A function \(f:\mathbb R\to\mathbb R\) satisfies \(f(xy)=xf(y)+yf(x)\) for all \(x,y\). Find \(f(0)\) and \(f(1)\).
First set \(x=0\), then \(x=1\).
For \(x=0\): \(f(0)=y f(0)\) for all \(y\). Taking \(y=2\), we get \(f(0)=2f(0)\), hence \(f(0)=0\).
For \(x=1\): \(f(y)=f(y)+y f(1)\). Taking \(y=1\), we get \(f(1)=f(1)+f(1)\), hence \(f(1)=0\).
Shows that \(0\) and \(1\) often give different information.