Let $f(z)$ be continuous on a region A and holomorphic on $A\setminus \{z_0\}$ for a point $\{z_0\}$. Show that f is holomorphism on $A$

Question

Let $f(z)$ be continuous on a region A and holomorphic on $A\setminus \{z_0\}$ for a point $\{z_0\}$. Show that $f$ is holomorphism on $A$

i really did't get any idea..can any help please

Answer 1

This is just a particular case of Riemann's theorem on removable singularities.

It's not hard to prove it. Let, for each $z\in A$, $g(z)=(z-z_0)f(z)$. Then $g$ is holomorphic, since$$g'(z)=\begin{cases}f(z)+(z-z_0)f'(z)&\text{ if }z\neq z_0\\f(z_0)&\text{ if }z=z_0.\end{cases}$$Besides, $g(z_0)=0$.

Since $g$ is holomorphic, it is analytic and, since $g(z_0)=0$, its Taylor series around $z_0$ is of the type$$g(z)=a_1(z-z_0)+a_2(z-z_0)^2+\cdots$$and therefore$$f(z)=a_1+a_2(z-z_0)+\cdots$$So, $f$ is analytic and therefore holomorphic.