Show that

(a) directly

(b) using the Cauchy Riemann equations

that if $\displaystyle f(z) \in \mathbb{R}$ $\displaystyle \forall z \in \mathbb{C}$, then for any $\displaystyle z \in \mathbb{C}$ if $\displaystyle f'(z)$ exists then $\displaystyle f'(z)=0$.

I don't know how to do this. For part (a) I have tried using the definition of the derivative and wasn't sure how to do it. I am not sure how to use the Cauchy Riemann equations, we have just started on them. Thanks for any help.