Continuity

Continuity is a concept denoting changes which do not happen very fast.

Objective Mathematics does not draw sharp distinctions between continuity and gradualness, nor does it draw sharp distinctions between discontinuity and abruptness. All four concepts are first-level concepts, that is, they directly identify percepts.

Whether a change is abrupt or gradual is a relative notion; a change may be identified as abrupt or gradual only when it is compared in consciousness to another change. In short, context matters. For example, in an everyday context, where one's point of comparison is days, hours, and minutes, the process by which a child grows into an adult is considered to be very gradual. By contrast, in a context where one is thinking about evolution by natural selection, i.e. where one's point of comparison is hundreds of millions of years, that same process is considered to be abrupt.

The traditional concept

What is the traditional concept of continuity?

It will be a long time before I write anything here. Go look it up. [TODO]

What is wrong with it?

The standard definition refers to things like real numbers and limits, concepts which must be treated with care if one wants to preserve one's connection with reality.

As a result of its malformed concepts, the standard definition of continuity is not as practical as Objective Mathematics' definition. For example, consider the following two functions:

$f(x)={\begin{cases}x,&{\text{ if }}x\geq 1\\0,&{\text{ otherwise}}\end{cases}}$,

$g(x)={\begin{cases}x,&{\text{ if }}x\geq 1\\(x-0.99999999999)/0.00000000001,&{\text{ if }}0.99999999999\leq x<1\\0,&{\text{ otherwise}}\end{cases}}$.

In practice, there are many contexts where it is impossible to distinguish between these two functions. For example, an engineer whose ruler only goes down to 0.01 (in the appropriate units) would be completely unable to distinguish between them. Despite that, the formal math definition disregards context and always makes a sharp distinction between these two functions: $g$ is continuous, and $f$ is not.

This example could be much more drastic, for example instead of contrasting $f$ with $g$, I could have contrasted it with $h$, where

$h(x)={\begin{cases}x,&{\text{ if }}x\geq 1\\(x-(1-\epsilon ))/\epsilon ,&{\text{ if }}(1-\epsilon )\leq x<1\\0,&{\text{ otherwise}}\end{cases}}$

and where $\epsilon$ is the reciprocal of 10^10^10^10^10. Quantities which are that small (in terms of non-pathological units) are very far outside the range of quantities that we expect might be measurable someday; it is physically impossible to distinguish between $f$ and $h$.

All of this proves that in real life, when given a function, it is never possible to tell if the function is continuous or not, according to the traditional definition of continuous. What's the point of having a concept if you can never actually use it to identify things in real life?

Analytic definition of continuity

Surely it must sometimes be useful to be more precise about continuity. I will show how Objective Mathematics would treat continuity in the case of a function, $f:\mathbb {Q} \rightarrow \mathbb {Q}$.

There must be some context. Something is changing fast

Geometric definition of continuity

[TODO it will be a long time before I write anything here]

Examples

For a perceptual example, contrast the following two images [TODO]

In comparison to (A), we identify that (B) undergoes an abrupt change of height; in comparison to (B), we judge that (A) undergoes a gradual change of height.

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Rubber sheet geometry [TODO]

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Knots [TODO]