Problem
ALG-B3-M01-P001 Shift of the argument
#1
★★☆☆☆ Level 2 of 5
Find all functions \(f:\mathbb R\to\mathbb R\) such that \(f(x+y)=f(x)+y\) for all \(x,y\in\mathbb R\).
Hint. Substitute \(x=0\).
With \(x=0\), \(f(y)=f(0)+y\). Hence \(f(x)=x+c\), where \(c=f(0)\). Verification: \(f(x+y)=x+y+c=f(x)+y\). Answer: \(f(x)=x+c\), \(c\in\mathbb R\).
Goal: separate guessing the answer from a complete proof and domain check.