While reading a book around complex analysis I came across a small theorem:
If $f(z)$ is a real valued, complex variabled, analytic function then it is constant.
I started thinking that if this is true then given any complex variabled, complex valued, analytic function $f$ we can see that $$\Re(f), \Im(f)$$
are both real valued, complex variabled, analytic functions thus must be constant and thus $f$ is constant.
Now I know that in reality not all complex valued, complex variabled, analytic functions are constant. But I want to know what is wrong with the "proof".