I was reading Complex Analysis By E Stein and Shakarchi.
In that Following 2 defination I had occured 
As $z_0$ single point which is also closed in $\mathbb C$
So by definition of Holomorphic on Closed set, It says that around some open containing $z_0$ is holomorphic (differentiable).
But I am not able to prove this form definition of Limit.
How to prove this ? or my observation is wrong.
ANy Help will be appreciated
Note first that these are not the standard definitions!
This set of definitions seems like a very bad idea, giving non-standard definitions for such basic concepts. It also seems like a bad idea regardless of whether it's standard, because the definition of "holomorphic at $z_0$" and "holomorphic on $\{z_0\}$" are not the same - that seems like they're trying to be confusing.
That last point resolves the confusion about the function $f(z)=|z|^2$: by these definitions it is holomorphic at $0$ but it is not holomorphic on $\{0\}$. No contradiction, but hideous terminology.
Here are the three corresponding standard definitions:
$f$ is differentiable at $z$ if $f$ is defined near $z$ and $f'(z)=\lim_{h\to0}((f(z+h)-f(z))/h$ exists.
If $\Omega$ is open and $f:\Omega\to\Bbb C$ then $f$ is holomorphic in $\Omega$ if $f$ is differentiable at $z$ for every $z\in\Omega$.
(In particular we only speak of "holomorphic" in an open set.)
If $E$ is any set and $f:E\to\Bbb C$ then $f$ is analytic on $E$ if $f$ can be extended to a function holomorphic in some open set containing $E$.
I can't imagine why S&S changed the standard terminology. The standard definitions seem much less likely to lead to the sort of confusion you express in your question.