Discontinuous Analysis

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 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).