# Complex function not continuous at $z_0$

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 $z_0$?

• No, because having a derivative there implies continuity there! – Shashi Jan 28 '18 at 21:09
• Ok thank you! I was not sure if this carried from real analysis – Abrb Jan 28 '18 at 21:11
• @Abrb Identify $\mathbb{C}$ with $\mathbb{R}^2$ math.stackexchange.com/questions/1187000/… – mysatellite Jan 28 '18 at 21:13

$\ f\ is\ analytic \ on \ region \ D \Rightarrow\ f \ is \ continuous \ on \ D$. Therefore $\ f$ cannot be analytic on D if there's a point in D in which $\ f$ is not continuous.
• if you take $f(z)=log(z)$ how would you prove that it doesn't satisfy CREs on the negative real axis? – Abrb Jan 28 '18 at 21:59