Problem
ALG-B3-M01-P004 Zero in the image
#4
★★★☆☆ Level 3 of 5
Find all \(f:\mathbb R\to\mathbb R\) such that \(f(x+y)=f(x)f(y)\) for all \(x,y\in\mathbb R\) and \(f(0)=0\).
Hint. Substitute \(y=0\).
With \(y=0\), \(f(x)=f(x)f(0)=0\). Hence \(f\equiv0\). This function indeed satisfies the equation.
Goal: separate guessing the answer from a complete proof and domain check.