Complex function not continuous at $z_0$

Question

If I have a function $f(z)$ defined on a domain $D$ in the complex plane that is not continuous at a point $z_0 \in D$, can $f$ be analytic in the region $D$? or I guess another way to phrase this would be can $f$ have a derivative at $z_0$?

Answer 1

$\ f\ is\ analytic \ on \ region \ D \Rightarrow\ f \ is \ continuous \ on \ D $. Therefore $\ f $ cannot be analytic on D if there's a point in D in which $\ f$ is not continuous.

Answer 2

Generally speaking, you can always rewrite f(z) as: u(x,y) + i*'v(x,y), and see (at any point z = x + iy you like) if: du/dx +idv/dx = du/dy - i*dv/dy. If yes then it is analytic at that point.