There is a question asking if the function $f$, which is defined by $$f(z) = \frac{\bar{z}^2}{z},\quad z\neq 0$$ and $f(0) = 0$, is analytic at $z = 0$ and if there is $f'(0)$.
The answer is supposed to be it is analytic since CR equations hold but the limit depends on a variable so there is no derivative at $0$. We have also been given theorems implying that analytic property is equivalent to having a derivative which is also equivalent to CR equations which is not true I think. In an exam we are asked to find the set in which the function is not analytic and in the answer sheet, only the points that the function was not continuous have been shown. So, I am very confused about it. What are the conditions for a function to be analytic? Are there any tests to find out if a function is analytic?