Question

The function f(z)=|z|^{2} is differentiable

Answer 1

$f(z)=|z|^{2}= z \bar{z}\\$

If f is differentiable at $z=z_0$

Then $\frac{\partial f}{\partial \bar{z}}|_{z=\bar{z}}=0\\$

Now

$\frac{\partial f}{\partial \bar{z}}=z\\ for \space z(\not=0) \in \nsubseteq \frac{\partial f}{\partial \bar{z}} \not=0\\ \implies f(z)=|z|^2$ is not differential at $z(\not=)=0 \in \nsubseteq$

To check at z=0

$f'(0)= \lim\limits_{z \to 0 }\frac{f(z)-f(0)}{z-0}\\ f'(0)= \lim\limits_{z \to 0 }\frac{z \bar{z}-0}{z}\\ f'(0)=0$

Hence f(z) is differentiable at z=0 only.