Problem
ALG-B3-M02-P002 Substitution of zero
#2
★★☆☆☆ Level 2 of 5
Prove that there is no function \(f:\mathbb R\to\mathbb R\) satisfying \(f(x+y)=f(x)+f(y)+x\) for all \(x,y\in\mathbb R\).
Hint. Take \(y=0\).
With \(y=0\), \(f(x)=f(x)+f(0)+x\), so \(f(0)=-x\) for all \(x\). This is impossible. Therefore there are no solutions.
Goal: choose the right first substitution and then verify the found function.