Q1: Which two properties does an ‘open domain’ as a subset of $\mathbb{C}$ fulfil?

A1: open and path-connected

A2: closed and path-connected

A3: closed and open

A4: open and no holes

Q2: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be holomorphic with the property that $$ \oint_{\gamma} f(z) , dz = 0 $$ for all closed curves $\gamma$. What is a correct conclusion?

A1: $f$ has an antiderivative.

A2: $f$ has no antiderivative.

A3: $f$ has exactly two antiderivatives.

A4: $f$ is not differentiable.