Example of Looman-Menchoff theorem
By Abigail Rogers •
I would like to know of one example of a function that strictly satisfies the Loomann-Menchoff theorem.
Namely, I want a complex function $f=u+iv$, defined on some $A \subseteq \mathbb C$, such that
- $f$ is holomorphic in $A$;
- $u$ and $v$ admit partial derivatives at each point of $A$;
- $u$ and $v$ satisfies the Cauchy-Riemann equations in $A$;
- $u$ and $v$ are continuous but not differentiable at some point of $A$.
For the life of me, I can not find such an example.
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