Available in the course:

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

  • It uses the theory of "funcoids", a highly abstract thing from 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.

  • Detailed article. There is a rather full definition and description of properties of generalized limit already. I am going to split it into smaller parts (while preserving the long artilce, too) and add definitions. Order the course now and receive both existing and ongoing study materials. More materials to be added, for example about derivatives.

  • Support from Newton, err... the author.

  • Course certification with simple exam questions will be added. Everybody who will pass receives "Master of discontinuous analysis" certificate!

  • In traditional calculus multivariable or infinite dimensional analysis is complex. In this course you get an easy way of it.

  • 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.

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

Course curriculum

  • 1

    Introductory video

    • Introductory video

  • 2

    Generalization of limit for an arbitrary function at every point

    • Generalization of limit for an arbitrary function at every point

FAQ

  • You are a crackpot! There is no limit of an arbitrary function at every point.

    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.

  • What are usages of discontinuous 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.

  • What are prerequisites?

    You need to know basic calculus, what is general topology, and what is linear algebra.

  • Are generalized derivatives and integrals linear functions?

    Yes!

  • To whiich kinds of numbers is it applicable?

    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.

  • Whom is this course for?

    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!