I'm looking for a function that satisfies Cauchy-Riemann equation on a whole domain but not differentiable. It seems there are some examples that satisfy Cauchy-Riemann equation on a point (usually the origin) but not differentiable at that point. Or, if the given function satisfies Cauchy-Riemann equation on a whole domain, then the function should be analytic?
Edit: The original question is that $\textit{a function f:G$\subset\Bbb C\to\Bbb C$ is analytic if and only if $\frac{\partial}{\partial\overline{z}}f = 0$}$. I found that $\frac{\partial}{\partial\overline{z}}f =0$ is equivalent to saying that $f$ satisfies the Cauchy-Riemann equation. How can I show that converse of the statement?
