You could simplify calculus

TL;DR

This paper presents a simplified, direct approach to calculus focusing on functions as primary objects, using division and Lipschitz conditions, avoiding traditional reliance on limits and continuity, and emphasizing computational methods.

Contribution

It introduces a bottom-up calculus framework based on uniform Lipschitz differentiability, providing elementary yet comprehensive tools for differentiability and monotonicity without heavy ontological assumptions.

Findings

Defines uniform Lipschitz differentiability (ULD) and shows its equivalence to divisibility in Lipschitz functions.

Recovers classical differentiability and monotonicity theorems through elementary estimates.

Extends the approach to functions of multiple variables briefly.

Abstract

I explain a direct approach to differentiation and integration. Instead of relying on the general notions of real numbers, limits and continuity, we treat functions as the primary objects of our theory, and view differentiation as division of f(x)-f(a) by x-a in a certain class of functions. When f is a polynomial the division can be carried out explicitly. To see why a polynomial with a positive derivative is increasing (the monotonicity theorem), we use the estimate |f(x)-f(a)-f'(a)(x-a)|<=K(x-a)^2. By making it into a definition we arrive at the notion of uniform Lipschitz differentiability (ULD), and see that the derivative of a ULD function is Lipschitz. Taking different moduli of continuity instead of |.|, we get different flavors of calculus, each rather elementary, but all together covering the total range of continuously differentiable functions. Using functions continuous at a, we recapture the classical definition of differentiability at a point. We also see that ULD is equivalent to divisibility of f(x)-f(a) by x-a in the class of Lipschitz functions of two variables, x and a. The analogous fact is true for any subadditive modulus of continuity. In this bottom-up, computational, one modulus of continuity at a time approach, the monotonicity theorem takes the central stage and provides the aspects of the subject important for practical applications. The weighty ontological issues of compactness and completeness can be treated lightly or postponed, since they are hardly used in this streamlined approach that pretty much follows the Vladimir Arnold's "principle of minimal generality, according to which every idea should first be understood in the simplest situation; only then can the method developed be applied to more complicated cases." I discuss a generalization to many variables briefly.