Suppose $f$ is analytic on $D-\{z_{0}\}$ and $f$ is continuous at $z_{0}$, then $f$ is analytic at $z_{0}$

Question

Let $D \subseteq \mathbb C$ be a domain containing a point $z_{0}$, $f$ be analytic on $D-\{z_{0}\}$ and $f$ is continuous at $z_{0}$, then Can we say that $f$ is analytic at $z_{0}$?

Answer 1

Yes.

To prove it, we can assume that $D$ is convex. You can show that the complex integral of $f$ along quadrilaterals not containing $z_0$ vanishes.

By using the continuity of $f$, deduce that the complex integral of $f$ along a triangle of which $z_0$ is a vertex vanishes.

Then, show that if $z_0$ belongs to the edge of a triangle, by splitting the triangle, the complex integral of $f$ along it vanishes.

By the same process, you can show that the integral of $f$ along any triangle vanishes. Then the standard argument shows that $f$ has a complex antiderivative, which is thus analytic, so $f$ is analytic.