I am a bit confused about differentiability in complex analysis. We showed in class that if a function $f$ is differentiable at $z_0$, then the Cauchy-Riemann equations hold at $z_0$. We also showed that if a function has continuous first partial derivatives at $z_0$, and the Cauchy-Riemann equations hold at $z_0$, then $f$ is differentiable at $z_0$. So apparently just satisfying Cauchy-Riemann at a point is not sufficient to determine differentiability.
So my question is: if one is given some function, what is a good way to see where this function is differentiable without going to the definition of the derivative?