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Lebesgue’s Road to Antiderivatives

Ádám Besenyei
Mathematics Magazine
Vol. 86, No. 4 (October 2013), pp. 255-260
DOI: 10.4169/math.mag.86.4.255
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Page Count: 6
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Lebesgue’s Road to Antiderivatives
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Summary The traditional way of proving the existence of antiderivatives of continuous functions is through the concept of definite integrals. In the years 1904–1905, H. Lebesgue provided an alternative proof of this result not relying on the theory of integrals. His method is based on piecewise linear approximations of continuous functions, which also yields the mean value inequality as a by-product. In this note we recall Lebesgue’s ideas.

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