Satisfying Cauchy Riemann equations at discontinuity What does it mean for a function to satisfy Cauchy-Riemann equations at a point discontinuity? The Cauchy-Riemann equations are about the partial derivatives of a function satisfying particular condition(s), but how would one calculate the partial derivatives at all at that point? Do you just take the limit of the partial derivative as one approaches that point?

Sastisfation of the Cauchy-Riemann equations alone is not sufficient to garantee the existence of $f'(z_0)$. For example: 
$f(z) = \begin{cases} \frac{\overline{z}^2}{z} &, z\neq 0 \\ 0 &,z=0 \end{cases}$
It satisfies the Cauchy-Riemann equations but $f'(0)$ does not exist. 
Now, if $f$ is defined in a certain neighborhood of $z_0$ and its first order derivative exists everywhere in that same neighborhood and the Cauchy-Riemann equations hold then $f'(z_0)$ exists. (see Chapter 2 - sec. 21 Churchill - Complex Variables and Applications).
I also suggest you that read a bit more about the concept of analytic function here.
