On Troikas: Disintermediation, Still Growth and the Tree

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.