a u(x,y) + b v(x,y) =c for all (x,y) $\in$ D then f is constant on D
2. But you don't know that f is constant, that's what you want to prove. So you might want to use the converse: if f'= 0 for all (x,y) then f is a constant. And you can do that using the Cauchy-Riemann equations to show that $\frac{\partial u}{\partial x}= \frac{\partial u}{\partial y}= \frac{\partial v}{\partial x}= \frac{\partial v}{\partial y}= 0$.