Editing Henry John Stephen Smith (section)

===Work on the Riemann integral===
In 1875 Smith published the important paper {{harv|Smith|1875}} on the integrability of [[discontinuous function]]s in [[Riemann integral|Riemann's sense]].<ref>See {{harv|Letta|1994|p=154}}.</ref> In this work, while giving a rigorous definition of the Riemann integral as well as explicit rigorous proofs of many of the results published by Riemann,<ref>The Riemann integral was introduced in Bernhard Riemann's paper "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe" (On the representability of a function by a trigonometric series), submitted to the University of Göttingen in 1854 as Riemann's ''Habilitationsschrift'' (qualification to become an instructor). It was published in 1868 in ''Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen'' (Proceedings of the Royal Philosophical Society at Göttingen), vol. 13, pages 87–132 (freely available on-line from [[Google Books]] [https://books.google.com/books?id=PDVFAAAAcAAJ&pg=RA1-PA87  here]): Riemann's definition of the integral is given in section 4, "Über der Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit" (On the concept of a definite integral and the extent of its validity), pp. 101–103, and {{harvtxt|Smith|1875|p=140}} analyzes this paper.</ref> he also gave an example of a [[meagre set]] which is not [[negligible set|negligible]] in the sense of [[measure theory]], since its measure is not zero:<ref name="Letta156">See {{harv|Letta|1994|p=156}}.</ref> a function which is everywhere continuous except on this set is not Riemann integrable. Smith's example shows that the proof of sufficient condition for the Riemann integrability of a discontinuous function given earlier by [[Hermann Hankel]] was incorrect and the result does not hold:<ref name="Letta156" /> however, his result remained unnoticed until much later, having no influence on successive developments.<ref>See {{harv|Letta|1994|p=157}}.</ref> In an 1875 paper, he discussed a nowhere-dense set of positive measure on the real line, an early version of the Cantor set, now known as the [[Smith–Volterra–Cantor set]].