I have read in Ahlfors’ complex analysis (p23):
“a real function of a complex variable either has the derivative zero, or else the derivative does not exist.”
and on p26:
“if $u$ and $v$ have continuous first order partial derivatives which satisfy the Cauchy Riemann equations, then $f(z)=u(z)+iv(z)$ is analytic with continuous derivative $f ’(z)$ and conversely.”
Now from Rudin’s book I know, that the derivative of a function g : ${\rm I\!R^n \longrightarrow \rm I\!R^m}$ exists and is continuous, if and only if the first order partial derivatives exist and are continuous.
Now my thoughts were: if $u$ and $v$ have continuous first order partial derivatives, it must follow from the latter theorem, that their derivatives $u’(z)$ and $v’(z)$ exist. But since they are both real valued functions of a complex variable, Ahlfors’ book would suggest that their derivatives must then be zero, which would mean that their partial derivatives are zero as well.
Now I know that this is wrong and I must have misunderstood something, but I don’t understand where I’ve made my mistake. Could someone please help me out?