# What's this rule called?

**URL:** <https://boards.straightdope.com/t/whats-this-rule-called/604069>\
**Category:** Factual Questions\
**Created:** [November 25, 2011, 6:37pm UTC](https://boards.straightdope.com/t/whats-this-rule-called/604069 "2011-11-25T18:37:01Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![Reyemile](https://avatars.discourse-cdn.com/v4/letter/r/46a35a/32.png) [@Reyemile](https://boards.straightdope.com/u/Reyemile)\
**Post date:** [November 25, 2011, 6:37pm UTC](https://boards.straightdope.com/t/whats-this-rule-called/604069/1 "2011-11-25T18:37:01Z")

</div>

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?

---

<div class="post-metadata">

**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\
**Post date:** [November 25, 2011, 6:49pm UTC](https://boards.straightdope.com/t/whats-this-rule-called/604069/2 "2011-11-25T18:49:30Z")

</div>

It’s a special case of the intermediate value theorem.

---

<div class="post-metadata">

**Author:** ![mnemosyne](https://avatars.discourse-cdn.com/v4/letter/m/c4cdca/32.png) [@mnemosyne](https://boards.straightdope.com/u/mnemosyne)\
**Post date:** [November 25, 2011, 6:51pm UTC](https://boards.straightdope.com/t/whats-this-rule-called/604069/3 "2011-11-25T18:51:58Z")

</div>

…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

---

<div class="post-metadata">

**Author:** ![Reyemile](https://avatars.discourse-cdn.com/v4/letter/r/46a35a/32.png) [@Reyemile](https://boards.straightdope.com/u/Reyemile)\
**Post date:** [November 25, 2011, 8:41pm UTC](https://boards.straightdope.com/t/whats-this-rule-called/604069/4 "2011-11-25T20:41:36Z")

</div>

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.