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

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

• What partial derivatives? What is the domain of the function you're looking for? May 24, 2019 at 3:42
• May 24, 2019 at 3:45
• Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. May 24, 2019 at 5:39
• @SeverinSchraven No, complex differentiability in an open set implies smoothness. May 24, 2019 at 15:55

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.