How do I properly punctuate bulleted list of mathematical equations? Here is an example:
Theorem 1 (Important Theorem). Consider f(x). If something, then

 1. The function has property A:

                  f(x) = g(x).

 2. The function has property B:

                  f(x) = h(x).

My question is the following:

do I capitalize the first letter of the bullet points?
do I include punctuation after the equation?

I typically aim to ensure mathematical expression follow the rules of grammar, but I am stuck here. If I don't capitalize the first letters, then I am not sure how to punctuate the equations.
I agree with you that mathematical formulas are usually sentences or parts of sentences and should be punctuated accordingly. This makes your example rather difficult if you don't want to change the words. I'd capitalize neither of the two bullet items, I'd put a comma after the first equation, and I'd put a period after the second equation.  That doesn't look very good, and besides it would really need &quot;and&quot; between the two bullet items.  So I suggest trying an alternative wording. Here's the first that occurs to me.
Assuming that the equations are intended to define (or summarize) properties A and B, a better solution would be to change the introductory &quot;If something, then&quot; to &quot;If something, then the function has the following properties.&quot; Then, instead of numbering the bullet items as 1 and 2, label them as A and B. (Even if A and B are words or phrases, you can use them as labels in LaTeX's &quot;description&quot; environment.) And put periods after both equations, since they're now stand-alone sentences rather than clauses in the &quot;If something ...&quot; sentence.