Question

Show that the function $f\left(z\right)=|z{|}^2$ for $z=x+iy$ is not differemtiable for $z\in C-\left\{0\right\}$.

Answer 1

Given the function $f\left(z\right)=|z{|}^2$.

or, $f\left(z\right)=x^2+y^2=u(x,y)+iv(x,y)$ (Let).

Then $u=x^2+y^2$ and $v=0$.

Now, $u_x=2x,u_y=2y,v_x=0$ and $v_y=0$.

So we have $u_x\ne v_y$ and $u_y\ne -v_x$ for $(x,y)\ne (0,0)$.

That is $f\left(z\right)$ doesn't satisfy Cauchy-Riemann equation for $z\in C-\left\{0\right\}$.

So the function is not differentiable for $z\in C-\left\{0\right\}$.