Complex differentiable function with non-continuous partial dervative [closed]

Question

I'm looking for a complex-valued function $f$ which is complex differentiable in $z_0$ but where the partial-derivatives are non-continuous in $z_0$.

Can someone give an example?

Best! Annette

Answer 1

We can cheat and modify the standard real-variable example: $$f(z)=\begin{cases}|z|^2\sin(1/|z|),&z\ne0, \\0,&z=0.\end{cases}$$. (To show $\partial f/\partial x$ is not continuous at the origin, note that $f(x)=x^2\sin(1/x)$ for $x>0$.)

Exercise Explain what would go wrong with the proof that $f'(0)=0$ if we'd said $z^2\sin(1/z)$ instead.

Exercise Show that the example modified as above is not differentiable at the origin.