Problem
ALG-B3-M01-P008 All additive shifts
#8
★★★★☆ Level 4 of 5
Find all \(f:\mathbb R\to\mathbb R\) such that \(f(x+y)=f(x)+f(y)+1\) for all \(x,y\in\mathbb R\).
Hint. Add \(1\) to the function.
Put \(A(x)=f(x)+1\). Then \(A(x+y)=A(x)+A(y)\). Thus all solutions are \(f(x)=A(x)-1\), where \(A:\mathbb R\to\mathbb R\) is any additive function. Verification by substitution is direct.
Goal: separate guessing the answer from a complete proof and domain check.