Related papers: Calculus without Limit Theory

We show that the instant motion of particle should be essentially discontinuous and random. This gives the logical basis of discontinuous motion. Since what quantum mechanics describes is the discontinuous motion of particles, this may also…

An alternative mathematics based on qualitative plurality of finiteness is developed to make non-standard mathematics independent of infinite set theory. The vague concept "accessibility" is used coherently within finite set theory whose…

This article shows a very elementary and straightforward proof of the Implicit Function Theorem for differentiable maps $F(x,y)$ defined on a finite-dimensional Euclidean space. There are no hypothesis on the continuity of the partial…

In calculus, an indefinite integral of a function $f$ is a differentiable function $F$ whose derivative is equal to $f$. In present paper, we generalize this notion of the indefinite integral from the ring of real functions to any ring. The…

In this paper, we investigate and find a necessary and sufficient condition for a function to be absolutely continuous over $\mathbb{R}$ (denoted by $AC(\mathbb{R})$) or any unbounded interval in $\mathbb{R}$ . Note that the Lebesgue's…

The traditional formulation of the ultimate goal of physics (in the narrower sense of axiomatic theory) involves the derivation of physical laws from first principles. Though, such option doesn't make things easier since the task of the…

Cirquent calculus is a new proof-theoretic and semantic framework, whose main distinguishing feature is being based on circuits, as opposed to the more traditional approaches that deal with tree-like objects such as formulas or sequents.…

The example of the calculus is used to explain how simple, practical math was made enormously complex by imposing on it the Western religiously-colored notion of mathematics as "perfect". We describe a pedagogical experiment to make math…

One of the main claims of the paper is that Dirac's calculus and broader theories of physics can be treated as theories written in the language of Continuous Logic. Establishing its true interpretation (model) is a model theory problem. The…

We introduce a non-wellfounded proof system for intuitionistic logic extended with inductive and co-inductive definitions, based on a syntax in which fixpoint formulas are annotated with explicit variables for ordinals. We explore the…

We report on a verification of the Fundamental Theorem of Algebra in ACL2(r). The proof consists of four parts. First, continuity for both complex-valued and real-valued functions of complex numbers is defined, and it is shown that…

We present a new way of organizing the few mathematical statements which form introduction to Calculus: the epsilon-delta characterization of the limit is now d e r i v e d from four simple, intuitive and frequently used statements, which…

This article has his origin in some lectures given at the University of Bologna, inside an interdisciplinary program of mathematics, history of science, physics and philosophy. Since they are at the junction of these fields, movement and…

Recently uncovered second derivative discontinuous solutions of the simplest linear ordinary differential equation define not only an nonstandard extension of the framework of the ordinary calculus, but also provide a dynamical…

We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using…

This paper presents a generalization for Differential and Integral Calculus. Just as the derivative is the instantaneous angular coefficient of the tangent line to a function, the generalized derivative is the instantaneous parameter value…

This is the English translation of my old paper 'Definici\'on y estudio de una funci\'on indefinidamente diferenciable de soporte compacto', Rev. Real Acad. Ciencias 76 (1982) 21-38. In it a function (essentially Fabius function) is defined…

This paper presents an algebraic-geometric construction of the derivative developed initially within the class of polynomial functions without introducing limits at the initial stage. Tangency is characterized by an algebraic condition: the…

We present a streamlined, slightly modified version, in the two-variable situation, of a beautiful, but not so well known, theory by B\"{o}gel, already from the 1930s, on an alternative higher dimensional calculus of real functions, a…

Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most…