This is a mathematical rule that’s pretty intuitive, and easy to prove:
If f(x) is a continuous from a to b, and if f(a) > 0 and f (b) < 0, then there exists c such that f© = 0.
Does this rule have a name?
It’s a special case of the intermediate value theorem.
…given that c is contained between a and b, of course.
This seems like a specific example of the Intermediate Value Theorem that states that if a real-valued function f is continuous on [a,b] and k is some number between f(a) and f(b), then there is some number c contained in [a,b] for which f© =k.
You’ve just specified a>0, b<0 and k =0
Just what I was looking for, thanks!
Just for the record, this came up in an internet debate about D&D. A poster insisted that one version of a power’s low damage made it too weak, while another versions damage was too high, and therefore the power was unfixable and couldn’t be properly balanced. This theory, which I was looking to cite, established that his position is contradictory; power as a direct function of damage is, for all intents and purposes*, continuous, so there must be a level at which it is balanced.
**it’s not truly continuous, because average damage is limited by the granularity of the dice, but if a fraction of a damage shouldn’t boost a power from ‘too weak’ to ‘too small.’ Also, if one damage is the difference between being able to one-shot a monster and being guaranteed to use two attacks, there is a very sharp increase in power (since overkill is wasted), but monsters have a wide enough range of hit point totals to make this largely moot.