Show that f(z)=conjugate(z) has no derivative

Question : Show that f(z)=conjugate(z) has no derivative

Solution :

We know that a complex conjugate finction is defined as

f(z)= z ̅=x-iy

Then  Re f(z) = u(x, y) = x and Im f(z) = v(x, y) = −y

Calculating the partial derivatives of u and v

∂u/∂x=1,   ∂v/∂y=-1 ,  ∂u/∂y=0,  ∂v/∂x=0

Since the Cauchy-Riemann conditions  

∂u/∂x=  ∂v/∂y  ,  ∂u/∂y  = -∂v/∂x  

are not satisfied at any point (x0, y0).

 Therefore f(z) = z ̅ has no derivative.