Conformal mappings - need explanation

[zezima1]

My book goes through this, but there are some things that I don't understand.
Let z = h(w), where h is analytic. Then F(w) = f(h(w)) is also analytic if f(z) is analytic. Thus in the z and w plane f and F satisfy the Laplace equation.
My book then says: "Further, suppose that Re f(z) is constant over a boundary C in the z-plane, then Re F(w) is constant over the boundary C' in the w-plane, where C' is the curve that C is transformed to in the w-plane.."
I have emphasized the last bit, because this is the thing I don't understand. What allowed you to make the conclusion that Re F(w) is constant over the boundary C' in the w-plane? Certainly the Re F(w) can also be a function of I am f(z), right?

 

[tt2348]

Well you have a curve C on the z-plane where Re f(C) is constant, the we also have C' which is the preimage of C in the w-plane , or C'=h^(-1) (C). Thus f(h(C')) =f(C), constant real part
Strictly on the boundary ofcourse