How to determine where a function is complex differentiable I know the definition of complex differentiability and also am aware that $f$ is complex differentiable at $z_0$ iff it is real differentiable at $z_0$ and that the partial derivatives satisfy the Cauchy-Riemann Equation.
In general, what are the steps required to show that $f$ is (not) complex differentiable at a point $z\in\mathbb C$?
EDIT: Made a typo on $f(z)$.
I was also given an example of a $f(z)=exp(-z^{-4})$ to show that $f$ satisfies the Cauchy Riemann equations at $0$ but is still not complex differentiable at $0$.
Does it mean that whenever we want to show whether $f$ is not differentiable at $z\in\mathbb C$, the Cauchy-Riemann Equation doesn't give us any information on it and we just have to work with the definition of complex differentiability and find sequences that tend to different limits?
 A: Two comments suffice for the specific function:

$f(z)=\exp(−z^4)$ is not even defined at $z=0$, so of course it does not satisfy CR in the first place. I assume that you considered the (non-continuous) extension obtained from additionally defining the value $f(0)=0$. 

– Hagen von Eitzen May 1 at 13:19

For this new $f(z)$, try the points $z_n=1/(e^{i\pi/4}n)$. Can the function be continuously extended at $0$? Not even speaking of differentiability. 

– julien May 1 at 13:24

As for the general question: 

Does it mean that whenever we want to show whether $f$ is not differentiable at $z\in \mathbb C$, the Cauchy-Riemann Equation doesn't give us any information on it and we just have to work with the definition of complex differentiability and find sequences that tend to different limits?

That depends on what the function is. If you are given $f(x+iy)=2x+iy$, you'll naturally point to the failure of the  Cauchy-Riemann equations. Working with difference quotients $(f(z)-f(z))/(z-a)$, while possible, would be unnecessarily awkward. 
When you inspect a function, you should be able to tell if it looks okay in the real sense, or if there are some quirks. If it looks real differentiable, then the CR equations are the thing to consider. Otherwise you try to disprove real differentiability in some way. 
