The Generalized Parallelism Theorem for Real and Vector-Valued Functions

A Geometric and Functional Approach to Non-Intersection Conditions

Abstract

This work introduces a new formalization of the concept of functional parallelism between real-valued functions, extending the classical meaning of parallel lines in the plane to generic continuous functions over a common domain. A theorem, referred to as the generalized parallelism theorem, is proposed to characterize the conditions under which two functions can be considered parallel in an expanded geometric sense. The rigorous definition is based on vertical translations and demonstrates that such a relationship implies the non-intersection of the respective graphs. This perspective unifies and formalizes an idea often implicit in the treatment of geometric transformations, offering an elegant and mathematically consistent generalization of traditional parallelism (1, 2).