to show whether a complex function is constant given some boundary conditions If a function is analytic on $B(0,1)$ and continuous as well as real valued on the boundary (i.e. $|z| = 1$),then show that the function is constant.

You may also use Schwarz' reflection principle and define for $|z|>1$:
$$ f(z)= \overline{f\left(\frac{1}{\overline{z}} \right)} $$
You may show that this extends $f$ to a uniformly bounded holomorphic function and apply Liouville.
