Suppose $f$ is a real-valued function of a complex variable that is differentiable at every $z \in \mathbb{C}$. Show that $f'(z)=0$ for all $z \in \mathbb{C}.$
My approach: Since $f$ is a real-valued function of a complex variable that is differentiable at all $z \in \mathbb{C}$, we can write that: $$f'(z)=\lim_{\lambda\to0} \frac{f(z+\lambda)-f(z)}{\lambda}=
L$$ for some real function $L$.
If this is true, then $f(z+\lambda) – f(z) = \lambda L + g(\lambda)$ such that $\lim_{\lambda\to0} \frac{g(\lambda)}{\lambda}=0$, where $g(\lambda)$ is a real valued function. However, if L is real, that implies $\lambda L$ is complex and arises from the subtraction of two real-valued functions. Since this is not possible, $\lambda L$ has to be real, which implies $L = 0$ or $L=c\bar{\lambda}$, where $c$ is some constant. But since L is real, L cannot be $\bar{\lambda}$. Hence, L has to be zero. This implies $f'(z) = 0$.
Is this proof correct? If not, how should I correct it?
Best Answer
The problem with your argument is that you have no control over $g $. Also, "complex" does not preclude "real".
Fix $z $. Since $f'(z) =L$ exists, we may approach along any line. So $$L=\lim_{t\to0}\frac {f (z+t)-f (z)}{t}\in\mathbb R. $$ Also, $$L=\lim_{t\to0}\frac {f (z+it)-f (z)}{it}\in i\mathbb R. $$ So $L $ is both real and imaginary: $L=0$.
As $z $ was arbitrary, $f'(z)=0$ for all $z $.