TITLE: Intuitive explanation why complex differentiation is not linear over real and imaginary part
QUESTION [7 upvotes]: Suppose we have a complex valued function $f = u + iv \colon U \to \mathbb C$ defined on an open subset $U$ of $\mathbb C$, which is holomorphic.
I was asked why it is wrong to use the linearity of the differentiation operator to write
$$
\frac{d}{dz} f(z) = \frac{d}{dz} (u(z) + iv(z)) = \frac{d}{dz} u(z) + i \frac{d}{dz} v(z).
$$
I explained this by proving that a real - valued function that is holomorphic must be constant, and so for $\frac{d}{dz} u$ and $\frac{d}{dz} v$ to exist $u$ and $v$ must be constant in $z$. However $f$ is not necessarily constant so applying the linearity in this way is invalid. I also showed that the real - and imaginary part functions $\Re()$ and $\Im()$ are not holomorphic.
Intuitively I think that real - valued and complex - valued functions are a different 'thing' and the $\frac{d}{dz}$ operator lives in the 'complex analysis world', so even though it is linear over complex functions it does not make sense to use it in the context of real - valued functions. However I was wondering whether there is a clearer or perhaps deeper explanation to this.

REPLY [3 votes]: The real and imaginary part are not analytic. The local Taylor expansion is
$u(z + h) = u(z) + ah + b\bar{h} + $ [sum of higher-degree polynomial functions of $(h,\bar{h})$]. 
The degree-1 approximation of $u(z)$ near $z_0$ is thus not of the form $u(z_0) +B (\Delta z)$ but $u(z_0) + B (\Delta z) + C(\overline{\Delta z})$.  
In the typical situation where $C \neq 0$, the difference quotient $\frac{u(z + h) - u(z)}{h}$ will not converge to a limit as $h \to 0$, since it equals $(B+C\frac{\overline{h}}{h}+o(1))$ which converges only along (asymptotically) radial paths.
So the perturbation with respect to $z$ makes good sense, $du = u(z + dz) - u(z)$, but its description in terms of $dz$ is no longer consistent with the idea of approximate $\mathbb{C}$-linear dependence on $dz$ alone.  Rather, you need to introduce $d\overline{z}$ and the derivative operators dual to it and $dz$.