**Disintermediation, Still Growth and the Tree**

Consider this

**Disintermediation Trilemma:**
* The function f(x) is real and
continuous on the interval [a,b];

* f(x) equals zero nowhere on
interval [a,b];

* f(a) and f(b) are of opposite
sign.

Two of the trilemma imply the
negation of the third, by the Intermediate Value Theorem:

If f is continuous and never
zero, then it does not change sign;

If f is never zero but changes
sign, then it is discontinuous;

If f changes sign and is
continuous, then it is somewhere zero.

Let each Stooge affirm all terms of
each of these implications, to make a troika supporting the trilemma.

A real continuous nonzero
sign-changing function is a curious beast; it’s a kind of continuous Heap,
changing sides without crossing the boundary. Therefore let us call such a
function a “

**Smuggler**”, and this the**Smuggler Trilemma**.
Consider the tree. First it is less
than ten million microns tall, later it is more than ten million microns tall;
it grows continuously; yet you cannot find a microsecond in which it is

*exactly*ten million microns tall! So let us call this the**Paradox of the Tree.**
Now consider this

**Unmean Values Trilemma:**
* f(x) is differentiable on the
interval [a,b];

* f(a) < f(b) ;

* df/dx is positive nowhere on
[a,b].

You can also call this the

**Still Growth Trilemma**. Any two of the trilemma implies the negation of the third, by the Mean Values Theorem. As ever, this unpacks into three deduction rules, and expands into a troika.
Consider the tree. It increases
in size, at a continuous growth rate of zero. That is the

**Second Paradox of the Tree.**
## No comments:

## Post a Comment