if f(z) is analytic/holomorphic in a Domain D, prove f(z) must be constant in that domain if any of these are also constant in D:

Im(f(z)), Re(f(z)), |f(z)|

So, f(x + iy) = u(x, y) + iv(x, y)

u, and v have first partial derivatives and satisfy Cauchy-Riemann equations:

$\displaystyle \frac{du}{dx} = \frac{dv}{dy}$

$\displaystyle \frac{du}{dy} = -\frac{dv}{dx}$

How can you go about showing this? Thanks.