$f$ and $|f|$ are analytic.

Question

If $f$ and $|f|$ are analytic on a domain(open and connected) $D \subset \mathbb{C}$ , then show $f$ is constant on $D$. What I did: Since $|f|$ is analytic and since it is a real valued function then its derivative is zero in the domain and hence $|f|$ is constant. Thus we must have the derivative of $f$ is $0$ ( which I know how to prove). Thus $f$ must be constant. Is my logic correct? Thanks