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
Asked
Active
Viewed 115 times
0
-
What partial derivatives? What is the domain of the function you're looking for? – davidlowryduda May 24 '19 at 03:42
-
1See this https://math.stackexchange.com/questions/2822224/is-there-any-function-that-is-differentiable-at-z-0-but-not-analytic-at-the-same?rq=1 – Rohit Gupta May 24 '19 at 03:45
-
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. – dantopa May 24 '19 at 05:39
-
@SeverinSchraven No, complex differentiability in an open set implies smoothness. – David C. Ullrich May 24 '19 at 15:55
1 Answers
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.
David C. Ullrich
- 89,985