Necessary Aand Sufficient Characterization Of Absolutely Continuous Functions Defined Over Unbounded Intervals

TL;DR

This work extends the classical Lebesgue-based characterization of absolute continuity from bounded intervals to unbounded domains. By introducing the spaces $L^1_G(\mathbb{R})$ and $L^1_H(\mathbb{R})$, it proves that a locally absolutely continuous function is globally absolutely continuous on $\mathbb{R}$ exactly when its derivative lies in $L^1_G(\mathbb{R})$, establishing a precise, practical criterion for $AC(\mathbb{R})$. It further shows $L^1_G(\mathbb{R})\subset L^1_H(\mathbb{R})$ with proper inclusion, enriching the hierarchy of integrability spaces relevant to global AC on unbounded domains. A Venn diagram is provided to visualize these relationships with classical spaces like $L^1_{loc}(\mathbb{R})$, $L^1(\mathbb{R})$, and $L^{\infty}(\mathbb{R})$. These results sharpen our understanding of how derivative integrability governs global absolute continuity on unbounded intervals and offer more accessible criteria for verification.

Abstract

In this paper, we investigate and find a necessary and sufficient condition for a function to be absolutely continuous over $\mathbb{R}$ (denoted by $AC(\mathbb{R})$) or any unbounded interval in $\mathbb{R}$ . Note that the Lebesgue's Fundamental theorem of Calculus gives us a necessary and sufficient condition\cite{book:B} for a function defined over a closed interval [a,b] to be absolutely continuous ,and the condition is that the derivative of the function should be in $L^1_{loc}([a,b])$. However, we don't have any such sufficient condition on the derivative of a function that is absolutely continuous over unbounded intervals. One necessary condition is that the function must be locally absolutely continuous (denoted by $AC_{loc}(\mathbb{R})$), but it may not be globally absolutely continuous despite being locally absolutely continuous(we give an explicit example of this). \\ The theorem 1 in this paper gives us a necessary and sufficient condition for a function belonging in $AC_{loc}(\mathbb{R})$ to belong in $AC(\mathbb{R})$ in terms of its derivative and identifies the space to which the derivative of an $AC(\mathbb{R})$ function must belong to as $L^1_{G}(\mathbb{R})$ (a strict subspace of $L^1(\mathbb{R})$). \\ Moreover, we define a new space of functions called $L^1_{H}(\mathbb{R})$, and in theorem 2 we show that $L^1_G \subset L^1_H$, which helps us to find an easier criteria to check whether a function belonging to $AC_{loc}(\mathbb{R})$, belongs to $AC(\mathbb{R})$ or not. \\ Finally, we provide a Venn diagram to explicitly show the relation of the newly defined spaces $L^1_{G}(\mathbb{R})$ and $L^1_{H}(\mathbb{R})$ with respect to the spaces $L^1_{loc}(\mathbb{R})$, $L^1(\mathbb{R})$ and $L^\infty(\mathbb{R})$.

Theorems & Definitions (8)

• Definition 2.0.1: Absolutely Continuous Function
• Remark 2.0.6
• Example 3.0.1
• Theorem 3.0.2
• Remark 3.0.3
• Remark 3.0.4
• Theorem 3.0.5
• Remark 3.0.6