GATE Electronics and Communications (EC) 2019 Solved Paper

A function $\mathit{F}\left(\mathit{z}\right)$ is said to be analytic at a point $\mathit{z}=$ a then $\mathit{F}\left(\mathit{z}\right)$ has a derivative at $\mathit{z}=\mathit{a}$ and derivative exists at each neighbouring point of $\mathit{z}=\mathit{a}$ in domain $\mathit{D}$.
${\mathit{e}}^{\frac{1}{2}}$ at $\mathit{z}=0⟶{\mathit{e}}^{\mathit{\infty }}⟶$ No derivative
$\ln \mathit{z}$ at $\mathit{z}=0\to \ln \left(0\right)=-\mathit{\infty }\to$ does not exists
$\frac{1}{1-\mathit{z}}$ at $\mathit{z}=1\to \frac{1}{0}=\mathit{\infty }\to$ does not exists
But $\cos \mathit{z}$ exists for all values of $\mathit{z}$ so it is analytic over the entire complex plane.