## 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

### Keywords:

completeness; fundamental theorems calculus; Propp function

Zbl 1305.12002
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