Are there two conventional definitions of “holomorphic”? In Walter Rudin's Real and Complex Analysis, second edition, on page 213, two definitions are stated.  One of them says the derivative of $f$ at $z_0$ is
$$f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}.\tag 1$$
The other says $f$ is holomorphic in an open set if $f'(z_0)$ exists at every value of $z_0$ in that set.
By that definition, holomorphic doesn't just mean differentiable; it means differentiable at every point in some open set.
He never defines “holomorphic at” a point, but only “holomorphic in” an open set.
Before opening the book this afternoon, just remembering from years ago, I thought what it said was


*

*$f$ is differentiable at $z_0$ if $(1)$ holds; and

*$f$ is holomorphic at $z_0$ if the derivative exists at every point in some open neighborhood of $z_0$.


By that definition $f(z)=|z|^2$ would be differentiable at $0$ but not holomorphic at $0$.
Thus differentiability at an isolated point would be weaker than holomorphy at that point.
QUESTION: Do both conventions exist?
I take “$f$ is analytic at $z_0$” to mean $f$ has a convergent power series expansion at $z_0$, which actually converges to the right thing, the value of $f$, in the interior of its circle of convergence. Differentiability at every point in some open neighborhood of $z_0$ is enough to prove analyticity, but differentiability at $z_0$ is not. By one of the conventions above, one can say a function holomorphic at a point is analytic at that point; by the other one cannot.
 A: There is almost never any reason to talk about functions that are (complex-) differentiable at  isolated points (other than as exercises to show the students that the derivative may exist at some exceptional points, and to show that Cauchy-Riemann's equations don't quite tell the full story).
I can't think of any book or text that uses "holomorphic at a point", at least not meaning that $f'$ exists at that particular point. If I would see the term, I would interpret it as synonymous with "analytic at a point", i.e. holomorphic on some open neighbourhood of the point.
A: Yes, both conventions exist, and are, as far as I'm aware, standard. Differentiability is a pointwise property (in the sense that differentiablity at $x$ does not imply differentiability at any other point), and holomorphy (or analyticity) a local property (in the sense that holomorphy at $z_0$ implies holomorphy in a neighbourhood of $z_0$).
Fischer/Lieb, A Course in Complex Analysis, define

A complex-valued function $f$ defined on an open set $U \subset \mathbb{C}$ is (complex) differentiable at $z_0 \in \mathbb{C}$ if there exists a function $\Delta$ on $U$, continuous at $z_0$, such that $$f(z) = f(z_0) + \Delta(z)(z-z_0)\tag{1}$$ holds for all $z \in U$. If $f$ is complex differentiable at all points $z_0 \in U$, then we say that $f$ is holomorphic on $U$. We say that $f$ is holomorphic at $z_0$ if there exists an open neighbourhood of $z_0$ on which $f$ is holomorphic.

(The $z_0 \in \mathbb{C}$ is a typo, should of course read $z_0 \in U$.)
Ahlfors also differentiates between the derivative existing at a point and analyticity/holomorphy:

The class of analytic functions is formed by the complex functions of a complex variable which possess a derivative wherever the function is defined. The term holomorphic function is used with identical meaning.

A: I am writing this like a answer, but only because for me to add a comment here I have t have 50 reputation (but I don't have still).
So, Michael Hardy, I was with a similar doubt. Because in Churchill's Complex variables and its applications, there's a section where he define an analytical function just like a function which derivative exists in a point $z_{0}$ and in the points that are a neighborhood of the point $z_{0}$.
In the case of the function $f(z)={\vert}z{\vert}^{2}$, this function has a derivative defined in the point $z=0$, but in the $z=0$ neighborhood we cannot define the derivative (because the lateral limit does not exist, so the derivative does not exists too). 
But there is another case:  the function $g(z)={\frac{1}{z}}$. It's derivative is not defined for the point $z=0$, but it exists to the neighbor points of $z=0$. In this case, $z=0$ is a singular point of the function.
Note: I would like to write this like an comment (in fact, this is not the answer!), but my S.E. points are not enough to post a comment.
A: According to Wikipedia, a function $f$ is said to be holomorphic at a point $z_0$ if it is holomorphic in some neighbourhood of $z_0$. It goes on to say that $f$ is defined to be holomorphic on a non-empty set $A$ if it is holomorphic on some open neighbourhood of $A$.
I don't know if any other reference uses this terminology though.
