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**Dinkydoe** I have to show that: if $\displaystyle f$ and $\displaystyle \overline{f}$ are holomorphic on an open connected set $\displaystyle U$, then $\displaystyle f$ is identically constant.

We can write $\displaystyle f= u(x,y)+iv(x,y)$

With the Cauchy-Riemann equations is easily shown that $\displaystyle \frac{\partial u}{\partial x}=\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}=0$

However to show that $\displaystyle f$ is identically constant I don't see where I need to use that $\displaystyle U$ is a open set.

I understand that the condition "connected" is strictly necessary, otherwise f may take on different constant values on different components.

But does U necessarily need to be an open set. Is "containing an open ball" not enough?