Mathematics · Complex Analysis

For an analytic function f(z) on a domain D, which of the following statements are correct? (A) If the real part of f(z) is constant, then f(z) is a constant function. (B) If |f(z)| is a non-zero constant in D, then f(z) is a constant function in D. (C) If f'(z) = 0 everywhere in D, then f(z) is a constant function in D. (D) If |f(z)| is a non-zero constant in D, then f(z) is constant only for some z in D.