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The use of full covers in real analysis. (English) Zbl 0721.26003
The notion of a full cover of a bounded set in the reals was discussed by B. S. Thomson [Real Anal. Exch. 6, 77-93 (1981; Zbl 0459.26004)] but appears to go back to the late 19th century. Thomson’s lemma, which asserts that if \({\mathcal C}\) is a full cover of [a,b] then \({\mathcal C}\) contains a partition of [a,b], is the principal tool used in the present paper to prove various theorems in real analysis. Among the theorems proved: (1) A function whose lower derivative is nonnegative on an interval J is nondecreasing on J. (2) Every bounded a.e. continuous function on [a,b] is Riemann integrable on [a,b]. (3) If f is absolutely continuous on [a,b] and \(f'(x)=0\) a.e. then f is constant on [a,b]. (4) If f is continuous, g is nondecreasing, and \(| f'(x)| \leq g'(x)\) except for a countable subset of [a,b], then \(| f(b)-f(a)| \leq g(b)-g(a)\).

MSC:
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
26A46 Absolutely continuous real functions in one variable
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