A Polylogarithm Solution to the Epsilon--Delta Problem

TL;DR

This paper introduces a novel approach to the epsilon--delta relation in continuity, providing a polylogarithmic time algorithm to compute a special, invertible delta function that optimally pairs epsilon with delta.

Contribution

It presents a new theoretical framework for the epsilon--delta relation, including the existence of a unique, continuous, invertible delta function, and an efficient algorithm for its numerical computation.

Findings

The privileged delta function is continuous, invertible, and maximal.

The algorithm computes the delta function in polylogarithmic time.

Examples demonstrate the method's accuracy even without explicit formulas.

Abstract

Let $f$ be a continuous real function defined in a subset of the real line. The standard definition of continuity at a point $x$ allow us to correlate any given epsilon with a (possibly depending of $x$) delta value. This pairing is known as the epsilon--delta relation of $f$. In this work, we demonstrate the existence of a privileged choice of delta in the sense that it is continuous, invertible, maximal and it is the solution of a simple functional equation. We also introduce an algorithm that can be used to numerically calculate this map in polylogarithm time, proving the computability of the epsilon--delta relation. Finally, some examples are analyzed in order to showcase the accuracy and effectiveness of these methods, even when the explicit formula for the aforementioned privileged function is unknown due to the lack of analytical tools for solving the functional equation.