# Discontinuous Analysis

## Mastering Discontinuous Analysis: A Comprehensive Course for Mathematicians, Physicists, Economists, and Engineers

No limit of an arbitrary function? No root of -1? There is indeed a root of -1 and likewise there is a (generalized) limit of every function at every point, for all types of discontinuity in calculus. Learn to use it in practice for limits, sums, derivatives, integrals of arbitrary functions.

## For whom is discontinuous analysis?

Welcome to this thingy of non-continuous analysis: itās a calculus for mathematicians and advanced math for engineers course. This advanced calculus is just a must for a physicist. If you want to do calculus self-study, this course in functional analysis and its applications to differential equations is for you, too (this course is not in university programs). Because discrete calculus is of interest for software developers, I welcome programmers, too, to learn a sound foundation of discrete analysis.

You will learn advanced calculus tricks allowing to formalize such things as derivative of discontinuous function or sum of divergent series. In general this course generalized any kind of calculus limit from convergent only to any divergent case.

Discontinuous analysis allows to unite discrete calculus (discrete derivative, discrete integral) and the usual calculus into one powerful theory using general topology. My generalized derivative (not the same as generalized derivative in distributions theory) applies to *any* function whatsoever. The same applies to my generalized integral and generalized limit in general, as well as to divergent series math.

Derivative of non-continuous function is often not a number but an infinite dimensional vector. Nevertheless, because it is a vector in a linear space, formulas like f'(x) ā f'(x) = 0 still apply, what simplifies your calculations. You can for example add, substrate, etc. infinite series without worrying whether they are convergent. You can check convergence at the end of calculations.

I told that this is a must for physicist. Probably we will discover using this theory something new about black holes connecting general relativity and quantum mechanics into a new kind of quantum gravity theory ā share a Nobel Prize with me then. Anyway various topics about non-continuous analysis are perfect PhD research topics.

**Purchasing this book, you support carbon accounting and DeSci (decentralized science).**

## Available in the book:

We start with a graphical explanation and then go into mathematical details.

There are two definitions of generalized limit: an axiomatic one and a concrete one. In usual cases they are equivalent. Choose any of the two. You don't need to remember all the definitions when you just calculate, just follow simple rules.

We take limits on "filters". That's a way to describe limits, upper limits, lower limits, ..., gradients, etc. with the same simple formulas, rather than repeating similar definitions over and over as you see in calculus books. It's not hard to understand.

You learn some of general topology just by the way, without being taught it in a usual boring non-understandable way. You learn newest discoveries in general topology and it's easy.

Support from Newton, err... the author.

It uses the theory of "funcoids", a highly abstract thing from general topology. This course doesn't include detailed information on funcoids (just their definitions - several equivalent ones), so some proofs may be not understood by you (just believe me), but you can read about funcoids in freely available sources, if you want to check proofs.

Apply it not only to continuous functions but also to discrete analysis (read: graph theory). Yes, you can apply analysis to graphs. Software developers!

Video

1. Introduction

2. A popular explanation of generalized limit

3. Funcoids

4. Limit for funcoids

5. Axiomatic generalized limit

6. Generalized limit

7. Generalized limit vs axiomatic generalized limit

8. Operations on generalized limits

9. Equivalence of different generalized limits

10. Hierarchy of singularities

11. Funcoid of singularities

12. Funcoid of supersingularities

13. Derivative

14. The necessary condition for minimum

15. Example differential equation

My version of quantum gravity – research it and share Nobel prize with me

### Victor Porton

Your instructor is Victor Porton, the person who discovered *ordered
semigroup actions* (and wrote 500 pages about them), a theory as
general as group theory but unknown before. Victor Porton is a
programming languages polyglot, author of multitudinous softwares
and programming libraries, blockchain expert and winner of
multitudinous blockchain hackathons, author of several books,
a philosopher.

Victor Porton studied math in a university 4.5 years, but didn't receive a degree because of discrimination.

Discontinuous analysis is a generalization of analysis and functional analysis that studies both discontinuous and continuous functions using generalized limit (defined for every function at all points). This allows to define derivative and integral of every function and sum of any infinite series. Properties of generalized limits are studied using special spaces called funcoids.

No, my generalized limit isn’t the same as ultralimit and has better properties than ultralimits. My book takes a view on ultralimits that is novel and constitutes major new discoveries.

Of course, there is no such thing. But there is a generalized definition of limit that does exist for every function at every point. That’s similar to root of -1: it does not exist in real numbers but exists for a wider set, complex numbers.

Discrete analysis is modified mathematical analysis where for taking a derivative we use a discrete interval (such as neighbourhood nodes in a graph) as the difference instead of an infinitely small difference used in the usual math analysis.

Discontinuous analysis due to its universality allows to analyze both infinitely small and finite differences, so encompassing both the usual math analysis and discrete analysis.

In discontinuous analysis all series, derivatives, integrals, etc. exist. This allows for example to cancel f'(x) – f'(x) = 0 for every function without first checking that the derivative exists. That simplifies your work.

There is a conjectured quantum gravity theory based on discontinuous analysis.

You need to know basic calculus, basics of general topology, and basics of vector spaces. Of course you need basics of set theory and logic.

To follow all the proofs, you need first (partially) read this book.

It is applicable to functions on real numbers, complex numbers, vectors, infinite dimensional vectors, etc. The exact conditions are specified in the lessons, but it is applicable to a VERY broad class of spaces.

Mathematics for PhDs. Mathematicians! Engineers. Physicists! (If you are a physicist, you have no right not to take this course.) Economists? I think, economists. And you can apply this to graphs instead of continuous functions – software developers!

Functions between compact spaces are not yet well-researched in discontinuous analysis.

I remind that the theory of generalized functions orĀ distributionsĀ also allows to study functions with discontinuities and infinite values.

But to have for instance product of two generalized functions in terms of distributions, you need to check complex pre-conditions. In my theory, every algebraic operation defined on numbers is also defined for generalized “quantities”. So, you can freely multiple any two functions (if there is a multiplication in your space).

## Some things not ready

## State

My theory of quantum gravity is not checked for existence and uniqueness of of solutions. Solve it and share a Nobel Prize with me?

Discontinuous analysis on compacts is not yet described.