Question

Prove that the function $f\left(z\right)=\overline{z}$ for $z=x+iy$ is not differentiable for $z\in C$.

Answer 1

The function $f\left(z\right)=\overline{z}=x-iy=u(x,y)+iv(x,y)$(Let).

Then $u=x$ and $v=-y$.

Then $u_x=1,u_y=0$ and $v_x=0,v_y=-1$.

As $u_x\ne v_y$ although $u_y=-v_x$.

So the function does not satisfies Cauchy-Riemann equation.

So the function is not differentiable for $z\in C$.