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Integrable functions from Cauchy to Riemann and Darboux. (Fonctions intégrables de Cauchy à Riemann et Darboux.) (French)
Summary: The article gives a survey of integrability. The author considers the different types of integrals. First, he explains Cauchy’s work at the polytechnic school in 1823. Then he exposes Riemann’s attempt for integrating non-continuous functions, without anti-derivative. The author details Riemann’s example, consisting in the function $x\to f(x)=\sum\frac{(nx)}{n^2}$ where $(x)=x-p_x$, where $p_x$ denotes the integer closest to $x$.