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Would real analysis be complete without the fundamental theorem of calculus? (English) Zbl 1347.26007

In recent years, many results in real analysis have been shown to be equivalent to the completeness of \(\mathbb R\); see the paper by J. Propp [Am. Math. Mon. 120, No. 5, 392–408 (2013; Zbl 1305.12002)]. The present paper adds the fundamental theorem of calculus to this list of equivalencies. This is done by the following two theorems. (I) If \(\mathbb F\) is an ordered subfield of \(\mathbb R\), then \(\mathbb F\) is complete if and only if every continuous function on a closed bounded interval has a uniformly differentiable primitive. (II) If \(\mathbb F\) is an ordered subfield of \(\mathbb R\), then \(\mathbb F\) is complete if and only if every continuous function on a closed bounded interval is Riemann integrable. The main part of the proof is the construction of a continuous Propp function that has a primitive whose value at \(1\) is in \(\mathbb R\setminus \mathbb F\). This paper has several other interesting results related to this topic.

MSC:

26A06 One-variable calculus
26A03 Foundations: limits and generalizations, elementary topology of the line
26A42 Integrals of Riemann, Stieltjes and Lebesgue type

Citations:

Zbl 1305.12002
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