Let $z=x+iy$ and $f(z) = 3 x y^2 + i y^3$ let $\frac{\partial}{\overline{\partial z}} = \frac{1}{2} (\frac{\partial}{\partial x} + i \frac{\partial}{\partial y})$
Computing the Cauchy–Riemann equations of $f$ leads to $\frac{\partial}{\overline{\partial z}}f(z)=3ixy =0 \iff x=0 \text{ or }y=0$
Can we conclude that $f$ is holomorphic on the real and the imaginary line ?
I'm new to complex analysis I thought the domain where a function is holomorphic must be an open set.
Thank you for your help
Indeed. And therefore there is no non-empty open set $\subset\Bbb C$ such that the restriction of $f$ to $A$ is holomorphic.