TITLE: Holomorphic vs differentiable (in the real sense).
QUESTION [17 upvotes]: Why a holomorphic function is infinitely differentiable just because of satisfying the Cauchy Riemann equations, but on the other side, a two variable real function that is twice differentiable is not infinitely differentiable?
I'm asking this for two reasons:
1) $\mathbb{R}^2$ is somehow an analog to the complex plane (they are isomorphic).
2) Holomorphic means that is differentiable in any direction of the complex plane, so why if a function is differentiable in any direction of the real plane then it is not infinitely differentiable?

REPLY [5 votes]: The usual presentation of holomorphic functions (and complex analysis in general) is misleading; the analogue of a holomorphic function to a function on the real line is one whose derivative is zero!
Recall one nice property of holomorphic functions: if a point $p$ lies inside a region $D$, which has a boundary $C$, then you can find the value of a holomorphic function $f(p)$ by only a contour integral about the path $C$.
The analogue of this property to the real line is as follows: let there be a region (an interval) $D$, whose boundary is two points $C_+$ and $C_-$.  Then the value of a function at a point $p \in D$ is characterized entirely by the value of the function at $C_+$ and $C_-$.  The only way this is possible is if $f$ is constant on $D$ and has identical values at $C_+$ and $C_-$.
(Note: it's possible that $f$ has a delta function source in the region instead, but this is just the analogue of a meromorphic function.)
This generalization--functions whose derivatives are zero--is what is ultimately more useful when you start looking at vector calculus and higher-dimensional real spaces.  Scalar fields that obey $\nabla \varphi = 0$ obey the exact same property of being fully characterized by their boundary values.
Similarly, the correct analogue on $\mathbb R^2$ is for a vector field $F$ to have vanishing divergence and curl: $\nabla \cdot F = \nabla \times F = 0$.  Such vector fields also obey the same basic property of holomorphic functions.