Would Real Analysis be complete without the Fundamental Theorem of Calculus?

TL;DR

This paper explores the deep connection between the completeness of real numbers and key theorems in real analysis, notably including the Fundamental Theorem of Calculus, by examining uniformly differentiable anti-derivatives.

Contribution

It demonstrates that the Fundamental Theorem of Calculus can be characterized as equivalent to completeness when considering uniformly differentiable anti-derivatives, expanding the understanding of this fundamental link.

Findings

The FTC can be included in the list of completeness characterizations.

Uniformly differentiable functions provide new insights into completeness.

The second part of the FTC, the Evaluation Theorem, is discussed in this context.

Abstract

The paper continues the intriguing theme that many key facts of (single-variable) Real Analysis are not only crucially dependent on the completeness of the real numbers, but are actually equivalent to it. The list of these characterizations of completeness is long and contains many prominent items, but so far the "biggest price", the Fundamental Theorem of Calculus (FTC), had resisted inclusion in the list. We show that the FTC $can$ be included, if one considers uniformly differentiable anti-derivatives. In the process, we exhibit some interesting facts about uniformly differentiable functions, including an additional characterization of completeness. We also discuss the second part of the FTC, the "Evaluation Theorem".