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.
Best Answer
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.