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A Classification of Elements of Function Space F ( R , R )

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  • Mohsen Soltanifar

    (Analytics Division, College of Professional Studies, Northeastern University, 1400-410 West Georgia Street, Vancouver, BC V6B 1Z3, Canada
    Biostatistics Division, Dalla Lana School of Public Health, University of Toronto, 620-155 College Street, Toronto, ON M5T 3M7, Canada
    Biostatistics & Programming Division, Biometrics Department, ClinChoice Inc., 750-2 Robert Speck Parkway, Mississauga, ON L4Z 1H8, Canada)

Abstract

In this paper, we present a constructive description of the function space of all real-valued functions on R ( F ( R , R ) ) by presenting a partition of it into 28 distinct blocks and a closed-form formula for the representative function of each of them. Each block contains elements that share common features in terms of the cardinality of their sets of continuity and differentiability. Alongside this classification, we introduce the concept of the Connection , which reveals a special relationship structure between the well-known representatives of four of the blocks: the Cantor function, the Dirichlet function, the Thomae function, and the Weierstrass function. Despite the significance of this field, several perspectives remain unexplored.

Suggested Citation

  • Mohsen Soltanifar, 2023. "A Classification of Elements of Function Space F ( R , R )," Mathematics, MDPI, vol. 11(17), pages 1-10, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3715-:d:1228021
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