Let f be an analytic function at the open unit circle and continous at |z|=1.

Prove: If f(z)=1 on the upper half of the unit circle

( for $\displaystyle z=e^{i\gamma} $ where $\displaystyle 0<= \gamma <=\pi $

then f is a constant at the unit circle...

I've no idea about this question.., I'll be delighted to get some guidance....

Thanks !