Points where this function is continuous: $f(z) = 3/z$ for $z=\pm 1$, and $(z^3-1)/(z^2-1)$ otherwise 
In what points is this function continuous?
  $$f(z)= \begin{cases}
             \dfrac{3}{z} &\text{if}\;z=\pm 1 \\[6pt]
             \dfrac{z^3-1}{z^2-1} &\text{if}\;z\not=\pm 1\\
             \end{cases}$$

My attempt:
I think i need verify if $f$ satisfies Cauchy-Riemann equations, in other words: $u_x=v_y$ and $u_y=-v_x$
And see if the partial derivatives of $u,v$ are continuous.
But i'm stuck. Can someone help me?

You just need to go back to the very friendly continuity criterion check that $\lim_{z\rightarrow 1}f(z)=f(1)$, $\lim_{z\rightarrow -1}f(z)=f(-1)$ are true or not.
In the later case, the limit even blows up. The discontinuities are $z=1,-1$.
