I'm trying to find all continuously differentiable functions $f : \mathbb{C} \to \mathbb{C} $ satisfying the equation :
$\frac{\partial f}{\partial \bar{z}} = z $ on $ \mathbb{C}$.
The left hand side is the standard partial differential operator for complex functions:
$\frac{\partial f}{\partial \bar{z}} = \frac{1}{2} (\frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y} )$
I'm stuck because surely if $f$ is continuously differentiable it satisfies the Cauchy Riemann equations and since the CR equations imply that:
$\frac{\partial f}{\partial \bar{z}} = 0 \implies z = 0 $.
And so there are no such continuously differentiable functions ?
I know this is incorrect but I'm not sure why.
No, it is not incorrect. If $f$ is differentiable and its domain is an open subset of $\mathbb C$, then $\frac{\partial f}{\partial\overline z}$ is $0$ everywhere.