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In (Biacino 2018) the evolution of the concept of real function of a real variable at the beginning of 1900 is outlined, reporting the discussions and the polemics, in which some young French mathematicians of those years as Baire, Borel and Lebesgue were involved, about what had to be considered a genuine real function. In this paper, I consider in particular the contribution to real analysis theory done by some Italian mathematicians as Volterra, Peano, Ascoli, Arzela, etc., in the last decades of nineteenth century before the introduction of measure and integration theory by Lebesgue.

Contribution of Italian Mathematicians to Real Analysis in the last Decades of Nineteenth Century