Calculus discontinuity fixed! Analysis of discontinuous functions.
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: it’s 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) – f'(x) = 0 still apply, what simplifies your calculations. You can for example add, subtract, multiple, 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 – share a Nobel Prize with me then. Anyway various topics about non-continuous analysis are perfect PhD research topics.
Available in the course:
2. A popular explanation of generalized limit
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
14. The necessary condition for minimum
15. Example differential equation
My version of quantum gravity – research it and share Nobel prize with me
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.
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.
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.
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 or distributions 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).