Discontinuities in the identical satisfaction of equations

TL;DR

This paper introduces measures for the unavoidable discontinuities in operations satisfying a set of equations on metric spaces, providing evaluations in simple cases.

Contribution

It defines and evaluates new quantities that quantify the minimal discontinuities required for operations satisfying given equations on metric spaces.

Findings

Quantities measuring minimal discontinuities are introduced.

Evaluations are provided for simple cases.

The approach advances understanding of discontinuities in algebraic operations.

Abstract

For a metric space $(A,d)$, and a set $\Sigma$ of equations, some quantities are introduced that measure the size of discontinuities that must occur in operations satisfying $\Sigma$ (identically) on $A$. We are able to evaluate these quantities in a few easy cases.}