Class 11

Math

Algebra

Complex Number and Quadratic Equations

Show that the function $f(z)=∣z∣_{2}$ for $z=x+iy$ is not differemtiable for $z∈C−{0}$.

or, $f(z)=x_{2}+y_{2}=u(x,y)+iv(x,y)$ (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 $u_{x}=v_{y}$ and $u_{y}=−v_{x}$ for $(x,y)=(0,0)$.

That is $f(z)$ doesn't satisfy Cauchy-Riemann equation for $z∈C−{0}$.

So the function is not differentiable for $z∈C−{0}$.