A Popular Introduction into the Generalized Limit (of arbitrary discontinuous function)

Abstract. A review of my book “Generalized limit (of arbitrary discontinuous function)”. A popular introduction with graphs to the following topic:

I consider (a generalized) limit of an arbitrary (discontinuous) function, defined in terms of funcoids. The definition of the generalized limit makes it obvious to define such things as the derivative of an arbitrary function, integral of an arbitrary function, etc. My book also defines of non-differentiable solution of a (partial) differential equation. It’s raised the question of how do such solutions “look like” starting a possible big future research program.

Keywords: algebraic general topology, limit, funcoid, differential equations

A.M.S. subject classification: 54J05, 54A05, 54D99, 54E05, 54E17, 54E99

In this informal article, I will explain popularly my concept of generalized limit of an arbitrary (discontinuous) function. For details and proofs see [1] and [2].

For example, consider some real function f from the x-axis to the y-axis:

Take it’s an infinitely small fragment (in our example, an infinitely small interval for x around zero; see the actual book for an explanation what is infinitely small):

Next consider that with a value y replaced with an infinitely small interval like [y – ϵ; y + ϵ]:

Now we have “an infinitely thin and short strip”. In fact, it is the same as an “infinitely small rectangle” (Why? So infinitely small behave, it can be counter-intuitive, but if we consider the above meditations formally, we could get this result):

This infinitely small rectangle’s y position uniquely characterizes the limit of our function (in our example at x→0).

If we consider the set of all rectangles we obtain by shifting this rectangle by adding an arbitrary number to x, we get

Such sets one-to-one corresponds to the value of the limit of our function (at x→0): Knowing such the set, we can calculate the limit (take its arbitrary element and get its so to say y-limit point) and knowing the limit value (y), we could write down the definition of this set.

So we have a formula for generalized limit:

limxa f(x) = { ν∘f|Δ(a)rrG }.

where G is the group of all horizontal shifts of our space R, f|Δ(a) is the function f of which we are taking limit restricted to the infinitely small interval Δ(a) around the point a, ν∘ is “stretching” our function graph into the infinitely thin “strip” by applying a topological operation to it.

What all this (especially “infinitely small”) means? Read my writings such as [1].

Why do we consider all shifts of our infinitely small rectangle? To make the limit not dependent on the point a to which x tends. Otherwise, the limit would depend on the point a.

Note that for discontinuous functions elements of our set (our limit is a set) won’t be infinitely small “rectangles” (as on the pictures), but would “touch” more than just one y value.

The interesting thing here is that we can apply the above formula to every function: for example to a discontinuous function, Dirichlet function, unbounded function, unbounded and discontinuous at every point function, etc. In short, the generalized limit is defined for every function. We have a definition of limit for every function, not only a continuous function!

And it works not only for real numbers. It would work for example for any function between two topological vector spaces (a vector space with a topology).

Hurrah! Now we can define the derivative and integral of every function.

Now read [2] for a formal treatment of the above (even for multivalued functions and even wider cases) and also how to put a derivative of a non-differentiable function into a differential equation (that’s interesting, it builds an infinite hierarchy of infinities/singularities and defines topological structures on this hierarchy).

Bibliography

[1] Victor Porton. Algebraic General Topology. volume 1. edition 1. At https://mathematics21.org/algebraic-general-topology-and-math-synthesis/, 2019.

[2] Victor Porton. Limit of a discontinuous function. At https://mathematics21.org/limit-of-discontinuous-function/, 2019.

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