Determining whether functions are regular for $X = V(XY - WZ)$ I've been playing around with this object for a while and I'm trying to get used to proving whether functions are regular or not. 
For example, the function $f = \overline{w}/\overline{y}$ is certainly rational, but I'm not sure how to show that it isn't regular. 
On the other hand, I've found a function that I believe to be regular, $g = \overline{y}^2 + \overline{z}\overline{w}^2/\overline{y}$.
How can I prove these assertions? Presumably I want to use the relations on these variables given by $V(XY - WZ)$.  
 A: One nice way of doing this sort of thing is to note that, if the function could be extended to the entire variety, then in particular this can be done working over $\mathbb{C}$, and the resulting function would be holomorphic.  Then you can use some basic calculus (just look at a limit) to show that this isn't the case.
For instance, in the case you've described (which I'm assuming is an affine hypersurface rather than a projective surface, judging by your example functions), suppose there exists some regular function $f$ defined on the entire variety which agrees with the function $w/y$ whenever that function is defined.
Since $f$ is a regular function, it's locally equal to a rational function, and when we consider a rational function of a complex variable in the Euclidean topology, that function is continuous wherever it's defined.  So, in other words, that means that $f : X \to \mathbb{C}$ is continuous in the Euclidean topology.  In particular, for each $x_0 \in X$, the limit $\displaystyle\lim_{x \to x_0} f(x)$ must be defined.
So, what's the limit at $x_0 = (0, 0, 0, 0)$?  Well, if it exists, then the limit along any path to $(0,0,0,0)$ must be the same.  So consider the paths $\gamma_1(t) = (t, 0, t, 0)$ and $\gamma_2(t) = (0, 0, t, t)$, both of which clearly lie on your variety.  For $t \neq 0$, we have $f(\gamma_1(t)) = t/t = 1$ and $f(\gamma_2(t)) = 0/t = 0$.  So we must have both 
$$\displaystyle\lim_{x \to (0,0,0,0)} f(x) = 0 \text{ and } \displaystyle\lim_{x \to (0,0,0,0)} f(x) = 1.$$
