Envelope theorem for discrete choice sets? If we have a function
$$f(x)=\max_yg(x,y)$$

Then we can find the derivative $d/dx \ f(x)$ by realizing that 
$$(1): \quad \frac {\partial }{\partial y}g(x,y^*)=0$$
because of the first order condition for maximization. 

We can use this by recognizing that $$\frac d {dx}f(x)=\frac d {dx}g(x,y^*(x))=\frac \partial {\partial x}g(x,y^*)+\frac \partial {\partial y}g(x,y^*)\frac {dy^*(x)} {dx}=\frac \partial {\partial x}g(x,y^*)$$
Where the last equality follows because of result $(1)$. 



However, my question is, what if $y$ can only take a discrete number of values? such as $0$ or $1$? We can of course say: 

$$f'(x)=\begin{cases}
g_x(x,0) \quad \text { if } g(x,0) &gt; g(x,1)\\
g_x(x,1) \quad \text { if } g(x,0) &lt; g(x,1)
\end{cases}$$

However, I am wondering if there is a kind of "envelope theorem for discrete choice sets", that would allow us to simplify this (especially if the choice set is discrete but large).
There is an envelope theorem for the setting you describe.

Have a look at “Envelope Theorems for Arbitrary Choice Sets” by Milgrom and Segal (2002).