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Posolutely and Absitively: The Alternate Yet Universal Definition of the Absolute Value Function

Author:

Gilreath, William Fletcher

Category:

Research Papers

Sub-Category:

Mathematics and Applied Mathematics

Language:

English

Date Published:

December 20, 2025

File:

/Research Papers-Mathematics and Applied Mathematics/Download/10413

Downloads:

Keywords:

Absolute value function, composite function, continuous function, derivative, differential, differentiable function, function composition, modulus function, power function, sign function, sub-function

Abstract:

The absolute value function, also known as the modulus function, is a ubiquitous mathematical function in algebra. The absolute value function, in terms of mathematical mapping, is also simple. The absolute value function takes any real number and maps it to either zero or a positive real number. In practice, the absolute value function measures the distance of a real number from zero, which serves as the point of origin. However, the existing, traditional definition of the absolute value function is flawed. The flaw is in the derivative or differential for the real absolute value function; the absolute value function has a derivative for all real numbers except at zero. The absolute value function is not differentiable at zero. The derivative for all real numbers except zero is defined and given by a step function. Yet the real absolute value function is a continuous function that has the global minimum where the derivative does not exist and is not differentiable, and thus is discontinuous at zero. This is the flaw and defect of the traditional absolute value function. While this defect or flaw does not have an impact on the mathematical mapping, the defect does create an inconsistency and irregularity in the mathematical properties of the real absolute value function.

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