In chapter 2 of "Complex Analysis" by Lars V. Alfors, the author concluded that "a real function of a complex variable either has the derivative 0 or else the derivative does not exist."
$\displaystyle \lim_{h \to 0} \frac{f(x+h+iy)-f(x+iy)}{h}$ is a real.
$\displaystyle \lim_{k \to 0} \frac{f(x+i(y+k))-f(x+iy)}{ik}$ is a pure imaginary number.
If both are the same, it must be 0. I understand this argument, but I came up with a question.
If we write a complex function $f$ of a complex variable $z$ as a sum $f(z)=u(z)+iv(z)$ where $u,v$ are real function of a complex variable, the only possible derivative of $f$ would seem to be only 0 according to the above conclusion. I know that this is ridiculous and there are plenty of counter examples ($f(z)=z, f(z)=z^2,...$), but I cannot find out what's wrong with my argument.