CMI 2024 | Mathematics | Part A | Question: 2

Question

A region in $\mathbb{C}$ is a non-empty open connected set. Select all the statement(s) that are true.

• A. Let $f$ be a function on a region $\Omega$ such that the integral of $f$ along the boundary of any closed triangle in $\Omega$ is zero. Then $f$ is analytic on $\Omega$.
• B. There exist a region $\Omega$ containing the real interval $(0,1)$ and a non-zero analytic function $f: \Omega \rightarrow \mathbb{C}$ such that $f\left(\frac{1}{n}\right)=0$ for all positive integers $n$.
• C. Let $f$ be an analytic function on $\mathbb{C} \backslash\{0\}$ with an essential singularity at $z=0$. Then $\lim _{z \rightarrow 0}|f(z)|=\infty$.
• D. Every bounded analytic function on $\mathbb{C} \backslash\{0\}$ is constant.