Discussion
Let Ω be a non-empty open connected subset of \(\mathbb{C}\) and \(f: \Omega \to \mathbb{C}\) be a non-constant function. Let the functions \(f^{2}: \Omega \to \mathbb{C}\) and \(f^{3}: \Omega \to \mathbb{C}\) be defined by
\[ \begin{aligned} f^{2}(z) &= (f(z))^{2} \ \text{and} \\ f^{3}(z) &= (f(z))^{3}, \quad z \in \Omega. \end{aligned} \]
Consider the following two statements:
S1: If f is continuous in Ω and \(f^{2}\) is analytic in Ω, then f is analytic in Ω.
S2: If \(f^{2}\) and \(f^{3}\) are analytic in Ω, then f is analytic in Ω.
Then, which one of the following is correct?
Answer: C
Both S1 and S2 are TRUE.
S1: \(g = f^{2}\) is analytic and not identically zero (otherwise f would be 0, a constant). Its zeros are isolated. Near any point where \(g \neq 0\), the continuous function f is a continuous branch of \(\sqrt{g}\), which is analytic. At an isolated zero, f is continuous and analytic in a punctured neighbourhood, so by Riemann's removable singularity theorem it is analytic there too.
S2: where \(f^{2} \neq 0\), \(f = f^{3}/f^{2}\) is a quotient of analytic functions, hence analytic. At a zero of \(f^{2}\), f is 0 and \(|f| = |f^{2}|^{1/2}\) tends to 0, so f is bounded and continuous there, and the singularity of \(f^{3}/f^{2}\) is removable.
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