Let $S$ be the set of all functions $f: \mathbb{Z} \rightarrow \mathbb{C}$ such that $f(m+n)=f(m) f(n)$ for all $m, n \in \mathbb{Z}$ and $f(0)=1$. Then

• A

  there exists $f \in S$ such that $f(-1)=0$
• Correct answer
  B

  there exists $f \in S$ such that $f(1)=2+3 i$
• C

  there exists $f \in S$ such that $f(1)=0$
• D

  $f(1)=1$ for every $f$ in $S$

Hence, the correct answer is option: 'B'.