Given $f: A \subset \mathbb{C} \rightarrow \mathbb{C}$ a holomorphic function, I can represent the function as $f=u+iv, \ u,v:A \rightarrow \mathbb{R}$ and so $f$ can be seen as a function from $A' \subset \mathbb{R}^2$ to $\mathbb{R}^2$ under the identification of $\mathbb{C} \ni z=x+iy$ with $(x,y) \in \mathbb{R}^2$ and of $f$ with $F=(u,v)$. 

Now, the problem arise (for me) in two different direction, the first: 

A theorem in [Markushevich] ["Theory of functions of a complex variable"][1](http://rads.stackoverflow.com/amzn/click/082183780X) says that if we look at $f$ as $F$ (notation above) then $f$ is holomorphic if and only if $F$ is real differentiable and solve Cauchy-Riemann conditions. 

Now, this gives us a good method to say when $f$ is holomorphic (at least when we can manage somehow the differentiability of $F$), but (and so the book continues) there is a nice sufficent condition on $F$ which is the continuity of partial derivatives of $u(x,y)$ and $v(x,y)$ (plus C-R equations) to imply holomorphicity of $f$. And that's ok, but this is only a sufficent condition. [Wikipedia's page](https://en.wikipedia.org/wiki/Holomorphic_function) on holomorphic functions seems to do confusion: it says (in **"Properties"**)

> 
> If one identifies $\mathbb{C}$ with $\mathbb{R}^2$, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the Cauchy–Riemann equations, a set of two partial differential equations.
> 
> 
> 

Is it me, or this is wrong?

Second question: how the fact that $f$ is holomorphic, and so analytical, relates with $u(x,y)$ and $v(x,y)$? The problem, to me, arises when I try to show that a holomorphic function $f=u+iv$ solve the equation $\frac{\partial^2u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$. 

i) How can I say that the second (partial) derivatives of $u$ exists?
ii) What about the mixed (partial) derivatives of $u$ which are supposed to cancel? How the [symmetry requirements](https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives) on $u$ are satisfied?

In summary: which are some good characterisations of holomorphicity in terms of $u,v$?

Thank you in advance