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The following is a slightly edited version of my 31 January 2003 sci.math post archived at Math Forum. Riemann  introduced his integral in his December 1853 Habilitationsschrift thesis. In his thesis he also gave an example, correctly verified, of a Riemann integrable function whose discontinuities form a dense set. Riemann's thesis wasn't widely known ...

6

Peano did do something else, and not quite right, apparently. In 1969 Kennedy published a note in the American Mathematical Monthly titled Is There an Elementary Proof of Peano's Existence Theorem, where he wrote: "In 1886 Giuseppe Peano stated  that the initial value problem: $y' = f(x,y),\, y(a) = b$, has a solution on the sole condition that $f$ is ...

6

Cauchy made integral rigorous, and proved that integral (in the sense of Cauchy) exists for continuous functions. Riemann proposed a more general definition, (integral in the sense of Riemann) and introduced the new class of functions, which are called now Riemann-integrable. This class is strictly larger than continuous (or piecewise-continuous) functions. ...

6

It was known even before Newton and Leibniz that areas under curves can be found by inverting the "computation of derivatives" (drawing tangents). In explicit geometric form this "fundamental theorem of calculus" was derived by Newton's teacher Barrow, see Barrow's Fundamental Theorem by Wagner. Newton and Leibniz developed explicit symbolic methods for ...

4

There are errors in the assumptions of your question: That history about Hippasus of Metapontum is highly doubtful. There is the version that you mentioned. There is also the version according to which what he discovered (or revealed) was how to construct a dodecahedron. You claim that the Pythagoreans were able to find out whether any two ratios of ...

3

The reference to the book on wave propagation (Leçons sur la propagation des ondes et les équations de l'hydrodynamique. Cours du collège de France, Hermann, 1903) available at https://archive.org/details/leonssurlaprop00hada/page/n12 indeed contains this lemma in the Note I, on p. 352, under the form: if $p \in \mathbb{N}^*$ and \$ F \in \mathcal{C}^p (\...

2

Cauchy was indeed the first, although his version was weaker than the modern one. An intuitive geometric interpretation of the theorem, along with a proof motivated by it, is due to Bonnet. A very well sourced and illustrated account of history of the mean value theorem is A brief history of the mean value theorem by Besenyei, which describes many ...

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Newton proves the upper and lower sums converge when the function (actually curve) is positive and monotonic (decreasing) in Lemmas II, III in the Principia. Newton does not formulate the process in terms of integration per se, nor in terms of functions. It is a solely geometrical process. The proof is equally valid if the function is negative or ...

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