Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be a function. Which of the following statement(s) is/are true? (A) Consider $f$ as a function $(f_1, f_2) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose that for $i = 1, 2$, both $\frac{\partial f_i}{\partial X}$ and $\frac{\partial f_i}{\partial Y}$ exist and are continuous. Then $f$ is entire. (B) Assume that $f$ is entire and $|f(z)| < 1$ for all $z \in \mathbb{C}$. Then $f$ is constant. (C) Assume that $f$ is entire and $\operatorname{Im}(f(z)) > 0$ for all $z \in \mathbb{C}$. Then $f$ is constant.
Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be a function. Which of the following statement(s) is/are true?\\
(A) Consider $f$ as a function $(f_1, f_2) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose that for $i = 1, 2$, both $\frac{\partial f_i}{\partial X}$ and $\frac{\partial f_i}{\partial Y}$ exist and are continuous. Then $f$ is entire.\\
(B) Assume that $f$ is entire and $|f(z)| < 1$ for all $z \in \mathbb{C}$. Then $f$ is constant.\\
(C) Assume that $f$ is entire and $\operatorname{Im}(f(z)) > 0$ for all $z \in \mathbb{C}$. Then $f$ is constant.