Fractal Class-Preserving Limit: A Generalization of Classical Limits for Fractal and Non-Standard Domains

This paper introduces the concept of the <i>fractal class-preserving limit</i>—a novel generalization of the classical limit, defined over fractal or non-standard subsets of the real number line. Traditional limits often fail to describe function behavior on disconnected or highly irregular domains, such as the Cantor set. By restricting the limit process to a specific subset, the class-preserving limit provides a rigorous framework for analyzing continuity, convergence, and oscillatory behavior within such sets. Formal definitions, proofs, and illustrative examples (including continuous and discontinuous functions) are presented. The method is especially useful in mathematical analysis, differential equations, data science, biology, physics, and materials engineering, where classical assumptions about the domain may not hold. Potential extensions to topology and infinite-dimensional spaces are also discussed.The author acknowledges the use of OpenAI’s language model for assistance in drafting certain sections of this workl