Let $f$ be an analytic function such that $f(z)$ is an element of $\mathbb R$ for all $z$ element of $\mathbb C$. Prove $f$ is constant.
Here's what I have done -
$f(z) = c + i0$, where $c$ is an element of $\mathbb R$
So i have component functions
$u(x,y) = c$, $v(x,y) = 0$
The partial derivative $u_x = 0$ and the partial derivative $v_x = 0$
The derivative, $f'(z)$ = $u_x +i v_x$, so I have $f'(z) = 0$
As $f'(z) = 0$ the function must be constant.
Does that seem right? One thing that I noticed when looking at question is that if $f(z)$ is an element of $\mathbb R$ then it is automatically constant...isn't that correct? But I would expect they are looking for more than that in an exam situation...