Ahlfors' Differentiation

Question

In Complex Analysis, Ahlfors makes the claim that:

if $f(z)$ be real function of complex variable whose derivative exists at $z=a$ then the derivative is zero

I'm taking this as $f: \mathbb{C} \to \mathbb{R}$

His proof is straightforward since $$\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$$ must be real as $h \in \mathbb{R}$. But at the same time $$\lim_{h \to 0} \frac{f(a+ih) - f(a)}{ih}$$ must be purely imaginary. Therefore they must be equal (equal to $0$), or non existent.

So, my confusion arises when he proves the Cuachy-Riemann equations by writing $f(z) = u(z) + iv(z)$. I've looked at Lang's book as a reference, and we see that $u(z)$ and $v(z)$ are both real valued. But looking at the Ahlfors proof, how are they not both $0$ or non-existent?

Can anyone clear up my confusion? I'm not quite sure if I've read something wrong or otherwise.

Answer 1

Your confusion seems to be in making the assumption that since $f(z)$ is complex-differentiable, $u$ and $v$ must be. In fact Ahlfors' proof shows that the real and imaginary part of a complex-differentiable function are rarely themselves complex-differentiable.

Answer 2

The Cauchy-Riemann equations concern $u(x,y)$ and $v(x,y)$, which are real-valued functions of two real variables (with $f(z)=f(x+iy)=u(x,y)+iv(x,y)$).