A Classification of Elements of Function Space $F(\mathbb{R},\mathbb{R})$

TL;DR

This paper classifies the entire space of real-valued functions on R into 28 categories based on continuity and differentiability, and introduces a new concept called Connection linking four notable functions in real analysis.

Contribution

It provides a comprehensive classification of F(R, R) and introduces the Connection concept to reveal relationships among key functions in real analysis.

Findings

Classified F(R, R) into 28 distinct blocks.

Introduced the Connection concept linking four fundamental functions.

Highlighted unexplored perspectives in the field.

Abstract

In this paper, we classify the function space of all real-valued functions on R denoted as F(R, R) into 28 distinct blocks. 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 four wellknown real-valued functions in real analysis: the Cantor function, Dirichlet function, the Thomae function, and the Weierstrass function. Despite the significance of this field, several perspectives remain unexplored