[ianleeclark]

> We hold the demand curve constant for any such analysis.

And this is the portion that I'm critiquing. You're holding it as a constant curve sloping downwards.

> So yes, we do have an idea of some fundamental properties of the demand curve

Okay, sure, it relies on there being people who have demand for some commodity.

> it cannot simply be any polynomial

"Any polynomial" might be overselling it, but we can't assume it's sloping downwards.