Example of function $f\in C^{\infty}$ but not holomorphic

Question

I know that holomorphic function are infinitely differentiable .

I think converse not true .I am searching for counterexample. But I did not get .

Please can anyone suggest me how to find such example.

any help will be appreciated.

Answer 1

Complex differentiable (even once) implies analytic implies infinitely differentiable.

For (real) infinitely differentiable but not analytic, standard example is $e^{-\frac{1}{x^2}}$.