Differential and integral calculus are two of the most important mathematical discoveries of the last few centuries. Since the work of Isaac Newton in 1665, through Lagrange and Cauchy, to formalize this branch of mathematics—calculus has served a crucial role in most fields of human inquiry. Anyone who’s done even basic geometry and algebra would appreciate the importance of calculus in the natural sciences and the more sophisticated forms of physics.

The question for us is, at the moment, what exactly is calculus? You can further ask whether there are limitations to the techniques and how they can be improved if at all.

There are two primary forms of calculus—integral and differential calculus that are concerned with different mathematical problems. Differential calculus is associated with determining rates of change, playing a crucial role in continuity analyses while integral calculus helps determine size, volumes, and areas.

As I said before, differential calculus is used to calculate rates of change—this pertains to issues like calculating velocities, acceleration, and solving optimization problems. Using the principles of differential calculus, it’s possible for us to create predictive models for how certain variables will behave—the changes in variable values and so on.

Integral calculus is primarily used to calculate volumes and mass contained within physical spaces. The integration operation is also called the anti-derivative, used in mathematical analysis in tandem with differential calculus to solve mathematical problems.

One of the most important contributions of Algebraic General Topology is an alternative representation of continuity—a central concern in calculus. It further uses its representation of generalized limit to present a new version of differentiability that can be applied to a broader range of functions—even arbitrary discontinuous functions.

AGT moves away from infinitesimal calculus to use novel concepts like staroids, funcoids and, multifuncoids, to help pave the way for a new approach to mathematical analysis. While the fundamental ideas remain the same, my AGT is a much more powerful mathematical theory that encompasses mathematical concepts that are inexpressible in general or algebraic topology—as well as calculus.

If you’re interested in Algebraic General Topology and wish to see how it can help you become a better mathematician, you should read Algebraic General Topology book Volume 1. If you have any questions, you should get in touch with me today—my doors are always open for new debate on my math research ideas.

]]>Calculus has served a crucial role in most fields of human inquiry. Anyone who’s done even basic geometry and algebra would appreciate the importance of calculus. Learn more about calculus here:

]]>One of the fields of human inquiry that make extensive use of mathematics and calculus is economics. Contemporary economic theory relies heavily on mathematical concepts and ideas to make sense or and theorize about economic phenomenon to devise economic models/policies. Anyone who doesn’t know the basics of calculus and mathematical analysis can’t hope to become an economist—because there’s no way to conduct economic analysis without math.

The concepts of differentiability play a significant role in devising functions that explain economic behavior, identifying efficiency conditions, and other essential factors. My Algebraic General Topology, considering that its primary concern is continuity and differentiability, can very well change how we theorize about economic behavior altogether.

Econometrics is a field of statistical analysis that’s often used in economic experiments to create predictive models that might present accurate forecasts on economic behavior. Even at the most basic levels, students are taught the concepts of marginal utility, price elasticity, and optimization using ideas of the Lagrangian optimization function and vector calculus.

Underlying this use of mathematics is the idea that human behavior is easily modeled using principles of mathematics, but there’s a lot of grey area in this assumption. Regression analysis—the underlying mathematics in econometrics—assumes that human behavior is like any other variable that mathematics can explain. Critics of existing econometric techniques often say that the application of homogenous models across entire economic contexts—basically to whole societies—is faulty because they presume a causal pattern within any economic setting. A gaping loophole in this thought is the fact that existing mathematics is fully capable of expressing all the possible causal chains within the economic sphere of society.

The question here is one of possibility rather than of a critique—economic behavior is rational only in so far as it complies with the laws of calculus. But what if we do not fit mathematics into economic behavior—what if instead, we fit economic behavior into mathematical laws? How do the limits of mathematics determine what we understand about economic behavior?

If I were to consider the possibility that there is some economic behavior that isn’t expressible in existing mathematical models, would that behavior also be classified as irrational? That’s really a very flimsy standard to set for economic rationality.

Algebraic General Topology is a new field of mathematical analysis that presents a new expression for continuity and differentiability. When used correctly, it can even be used to describe limits for arbitrary discontinuous functions—some of which might also describe economic behavior.

The possibilities of AGT within the context of economics are boundless. With a new way of talking about mathematical concepts, we’ll have a new way of discussing economic behavior. If you’re interested in AGT, you should read my math research book titled Algebraic General Topology Volume 1. For any questions on my work, you should get in touch with me—I always welcome any questions that I can answer or entertain any criticism of my work.

]]>Topological continuity is a key concept within mathematical theory. Topological analysis has opened paths to a deeper understanding of continuity at a more abstract level. However, topological continuity isn’t the only form of continuity mathematicians study—discrete continuity is one type of continuity that is separate from topological continuity.

Algebraic General Topology is one way of bringing these different concepts of continuity under a single heading. Incidentally, AGT also describes continuity in the same way that is described to speak of discrete continuity.

We’ll need to look at some crucial concepts within topology and then apply these to certain sets to see how the general definition of topological continuity applies to specific sets. Or, see how we can generalize the definition of topological continuity from instances of continuous functions defined between any two spaces.

A topological space is a set X together with a topology T on it—that’s the basic definition of a topological space.

Before I go into greater detail about topological spaces, it’s important that I discuss what a topology is.

A topology T on a nonempty set X is a collection of subsets of X, called open sets, such that:

- The empty set and the set X are open.
- The union of an arbitrary collection of open subsets of X is open.
- The intersection of a finite collection of open subsets of X is open.

Speaking in general terms, the set X contains a topology T if X also contains collections of open sets that satisfy the above conditions. T is basically the subset of X which conforms to the above mentioned properties and if you can find sets like T in X, then X is a topological space.

The difference between two topological spaces depends on how we define open sets within the set. For example, we define open sets on R like (0,1)—this set contains all the values between 0 and 1, but not 0 and 1 themselves but this only applies to the real line and not other sets. A general topological definition of an open set is:

*A set S, such that every point in S has a neighborhood contained in S*

Consider the open set (0,1) in R. I can define a neighborhood around every point that is contained within it. The entire set of real numbers, incidentally, also qualifies as a topology.

Before I define topological continuity, I should mention that you can only define continuous functions between two topological spaces. The definition will make this clear as well:

*A function f: S→T between two topological spaces is continuous if the pre-image f*^{−1 }*(Q) of every open set Q**⊂**T is an open subset of S.*

As a function, f will map open sets of S onto T. If f happens to map all open sets in S specifically onto open sets in T, then f is considered continuous.

Discrete topologies are ones where every individual point represents an open set and is also a closed set. The set of integers on the real line, for example, is a discrete topology.

A continuous function from a discrete topology to another topological space—let’s say the real line—would be one that maps each member of the discrete topology onto an open set.

I felt that it was important to discuss continuity on discrete topologies to distinguish this idea that of discrete continuity. There are fundamental differences between continuity in discrete topologies and discrete continuity—discrete continuity is defined as:

a function *f* (on a set *U*) is a continuous function from *μ* to *ν* iff *f*∘*μ *⊆ *ν*∘*f*

Where *μ **and** ν *are directed graphs.

If you’re interested in topological spaces or continuous functions defined between different topological spaces and generalizations , you should read Algebraic General Topology Volume 1.

The book introduces my mathematical theory that generalizes limits across arbitrary discontinuous functions. As always, I’m open to new ideas and thoughts on my work and welcome open debate on any issue you find.

Learn more about pointless topology here.

]]>I remind that I defined generalized limit of arbitrary function. The limit may be an infinitely big value.

It allows to define derivative and integral of an arbitrary function.

I also defined what are solutions of partial differential equations where such infinities (instead of e.g. real numbers or complex numbers) are defined.

You may see in **my book** that for the simple differential equation y'(x) = -1/x^{2} we can consider either in a sense arbitrary generalized solutions or generalized solutions with a “pseudodifferentiable” derivative. The first one gives an arbitrary value in the zero point and the second a fixed real value in zero point. See the book for more details.

If we would instead consider the general relativity Einstein equations, we would probably get the following (not yet checked):

- We require the solutions to be pseudodifferentiable in time (or rather, timelike intervals).
- We do not require the solutions to be pseudodifferentiable in space(or rather, spacelike intervals).
- Then in the singularity point there would form during time of black hole forming a certain value with an infinite structure in the center which is determined by the values of the variables while the hole was forming but is
*not*a function of the characteristics of the already form black hole.

If that hypothesis is true, we have a solution to the black hole information paradox: the center of a black hole holds not a hole but an infinite value containing the information of how the hole was formed. This value is a constant, but does not depend on the “dead” form of an already formed black hole, rather it contains the information of how the hole was formed.

It’s a very interesting hypothesis. Just write a publication on this topic, unless I do it before you. Win a Nobel prize.

]]>We apply filters to existing sets to express otherwise inexpressible statements.They effectively allow us to refer to infinitely small or infinitely large sets and conduct mathematical analysis to develop valuable insights.

]]>I feel that continuity is best understood when we consider convergence at different levels of abstraction. While it’s fairly easy to understand the continuity of functions when they’re defined in spaces like R^{2}, with standards like:

- The left hand limit must equal the right hand limit.
- The function should have a finite value at each point throughout any given domain.

Definitions like the ones I’ve mentioned above are also called definitions of continuity derived out of point-wise convergence. In general, a point-wise convergence definition of continuity looks something like this:

There are other more rigorous definitions for continuity, one of the common definitions that’s taught in beginner level calculus classes is:

All the conceptual analysis set aside, there are problems with these point-wise definitions of continuity and convergence because they don’t preserve boundedness or the continuity of functions. Anyone who’s taken higher level functional analysis would know that there are much more general definitions of continuity, which aren’t counter-intuitive or downright false.

These are called uniform-convergence definitions of continuity.

To be precise, uniform convergence directly implies continuity of the function across any domain or subset of the domain. This definition doesn’t take into account convergence of the values of x, rather, it looks at the convergence of sequences of functions. Put simply, if you can show that the function in question belongs to a sequence of functions that converge uniformly across any domain, then the function is continuous across the entire domain as well. The formal definition is as follows:

**Definition A: **If a sequence {f_{n}} of continuous functions f_{n}: A→R converges uniformly on A⊂R to f: A→R, f is continuous on A.

As a method of proof, if you’re looking to prove that a certain function is continuous—all you need to prove is that the function in question belongs to a sequence of continuous functions on a domain A, which is a subset of the set of real numbers. Definition A, as I’ve written it above preserves the boundedness of sequences and preserves continuity in a way that’s not possible through the point-wise definition of convergence.

This definition A is a direct extension of Cauchy’s definition of uniform convergence, which is as follows:

**Cauchy’s Definition of Uniform Convergence: **A sequence {fn} of functions fn: A→ R is uniformly Cauchy on A if for every ε>0 there exists N∈N such that m, n>N implies that |fm(x) −fn(x)|<ε for all x∈ A.

To speak loosely, if the absolute difference between two successive functions belonging to the same sequence gets smaller, this implies that the sequence is converging. Assuming that such a convergence function exists, each function within the sequence is going to be continuous.

If you’re interested in studying continuity or the limits of functions, you should read Algebraic General Topology Volume 1. This book introduces my mathematical theory that generalizes limits across arbitrary discontinuous functions. As always, I’m open to new ideas and thoughts on my work and welcome open debate on any issue you find.

Learn more about my general topology book here.

]]>I’ve published a new edition of my book Algebraic General Topology. The new edition features “unfixed morphisms” a way to turn a category into a semigroup. (Certain additional structure on the category is needed.)

The book features a wide generalization of general topology done in an algebraic way. Now we can operate on general topological objects with algebraic operations.

The book features the concept of *funcoid* – something better than topological space. Topological spaces mostly become history, the new thing of general topology is funcoids. It also considers filters on the Cartesian product of sets, pointfree topology, “multidimensional” general topology, where traditional topological spaces are always two-dimensional in a sense, as they consider binary relations (a relation between one point and one set).

Completely overthrow your understanding of general topology with this book. The series of the book also presents even more abstract semigroup algebra of general topology and a generalization of limit for an arbitrary function.

]]>A discontinuity is any point on a function where one of the three possibilities arise:

- The right-side limit is unequal to left-side limit
- The function jumps suddenly
- The function goes to infinity at a certain point in the domain.

Continuity and limits, as understood in traditional calculus, rely on infinitesimally small sets to arrive at limits for arbitrary functions. This idea, when translated into other definitions of continuities, relies on ideas about convergence of sequences and the like.

Such an approach relegates analysis on discontinuous functions to the side-lines because these functions violate the principles of infinitesimal calculus and thus not worthy of consideration.

It’s understandable why mathematicians have given continuity and infinitesimally small sets this much import—calculations of a practical nature may seem much more relevant to our concerns. Even with these convictions, it doesn’t disqualify analysis of discontinuous functions from mathematical theorization.

One of the outcomes of Algebraic Generalized Topology is the generalization of limits of discontinuous functions. This is a relatively novel path, considering the extensive literature that relies on the fundamental theorem of calculus and the traditional understanding of continuity, but is no less relevant to mathematical analysis. It is a branch of mathematics that is capable of generalizing continuity as well as discontinuity across different mathematical spaces, effectively making mathematical analysis much more powerful than it is currently.

Funcoids and Reloids are the fundamental components of mathematical analysis in AGT. For the purposes of the reader, I think it would be prudent to go through the basic definitions. These include:

A relation from a set *A *to a set *B, *such that it is* *a quadruple (*A, B, α, β*) where *α *ϵ F(*B*)^{F}^{(}^{A}^{)},* α*ϵ F(*B*)^{F}^{(}^{A}^{) }such that:

∀*X*ϵF(*A*)*, Y*ϵF(*B*) : (Y∩*α*(X) ≠0 ⇔ *X*∩*β*(*Y*) ≠0)*.*

F(A) here denotes the set of filters on a set A.

A Reloid is defined as:

A Reloid from a set *A *to a set *B is *a triple (*A, B, F*) where,

F is a filter on the Cartesian product A x B of two sets

I intend to go into deeper explanation of my work in later publications, but, for perspective, I think it is also worthwhile to think of the definition of continuity in terms of Funcoids and Reloids. The definition is as follows:

* f ϵ *C(µ,ν) *f*** **∘µ ≤ ν∘ *f*

This definition applies to topological spaces and also to uniform spaces, proximity spaces, graphs, etc. I also take the concept of funcoids to create an elegant generalization of limits for arbitrary discontinuous functions.

I have also worked extensively with limits of discontinuous functions and mathematical nondifferentiable analysis in my math research monograph titled Algebraic General Topology. Anyone interested in my work should get in touch with me directly.

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