22 votes
Accepted

Who discovered smooth non-analytic functions of a real variable?

An excellent source for the early history of smooth non-analytic functions is the paper Gerald G. Bilodeau. The origin and early development of non-analytic infinitely differentiable functions, Arch. ...
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21 votes
Accepted

Why did mathematicians not see that $f_n(x)=x^n$ is a counterexample to Cauchy's "theorem" about limits of continuous functions?

"Counterexamples" to Cauchy's theorem were "discovered" as soon as he proved it. Of course Cauchy knew all these "counterexamples" but he insisted that his theorem and its proof is correct until his ...
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19 votes

What was the answer to this paradox before Cantor?

Zeno (around 500 BC) raised this paradox to argue against the notion of "plurality", arguing that a belief in the existence of many things rather than only one leads to absurd conclusions: If there ...
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13 votes

What was the answer to this paradox before Cantor?

Aristotle gave the first systematic rebuttal of Zeno, in particular he wrote in Physics: "…a line cannot be composed of points, the line being continuous and the point indivisible". According to ...
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13 votes

Why did mathematicians not see that $f_n(x)=x^n$ is a counterexample to Cauchy's "theorem" about limits of continuous functions?

I'll add to Alexandre's answer that with the way Cauchy thought about continuum and defined convergence, the "counterexamples" did not converge to the limits we ascribe to them today. To Cauchy "...
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13 votes

Who said $\pi$ is a constant since it is not even a real number?

This question is based on a misunderstanding. The statement that $\pi$ is constant has precise meaning: $\pi$ is a ratio of the length of circumference to the length of diameter. The statement that it ...
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13 votes
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What did Kurt Gödel mean by nonstandard analysis being "the analysis of the future"?

The quote is from the remark Gödel made after Robinson's talk at the Institute for Advanced Study in Princeton in March 1973. It is reproduced in the preface to the second edition of Robinson’s Non-...
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13 votes
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Riemann's Contribution to Integration

The following is a slightly edited version of my 31 January 2003 sci.math post archived at google groups. Riemann [6] introduced his integral in his December 1853 Habilitationsschrift thesis. In his ...
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12 votes

Was the Riemann Integral the first integration theory?

The first rigorous integration theory in due to Eudoxus and Archimedes. It is called the method of exhaustion, and it allowed Archimedes to find the volumes of the balls, pyramids, cones, areas of ...
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11 votes
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Who discovered the difference between the infinities?

Georg Cantor discovered it. You can see at least : The Early Development of Set Theory : in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence ...
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11 votes
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What were the earliest “unpleasant” consequences of the Axiom of Choice (and its negation) to be deduced?

Vitali's construction is probably the first example "unpleasant" in the modern sense, but well-ordering of the continuum was unpleasant enough to some at the time of Zermelo's proof in 1904. Accepting ...
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10 votes

Are there written (19th century) sources expressing the belief that the intermediate value property is equivalent to continuity?

One answer to your question could be that the separation actually came pretty late. Wikipedia claims that "Earlier authors held the result to be intuitively obvious, and requiring no proof.", so until ...
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  • 1,795
10 votes

Who attached Buniakovsky's name to the Cauchy-Schwarz inequality?

Some reflections of J. Michael Steele (cf. The Cauchy-Schwarz Master Class. Cambridge University Press, 2004, pp. 10-12) on this matter: THE PACE OF SCIENCE -- THE DEVELOPMENT OF EXTENSIONS ...
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10 votes
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When was the first recorded occurence of irrational and imaginary number usage in number theory?

Irrational numbers were used by the ancient Greeks when they were discovered. The earliest texts did not survive but there are plenty of them in Euclid. Though they are not called numbers. The theory ...
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9 votes

Who attached Buniakovsky's name to the Cauchy-Schwarz inequality?

Maybe it is interesting to note that the term "l’inégalité de Schwarz" was coined by Poincaré in an 1896 paper in Acta Mathematica 20, p. 73, and was used in the French and German literature for the ...
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9 votes
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Historical occurrences of mathematicians substituting terms for $x$ in the denominator of $\mathrm{d}y/\mathrm{d}x$?

Gibbs (1889, p. 140): $ \qquad \dfrac{d\,\log\mathrm V}{d\,\log p} = - \dfrac{d\,\log n}{d\,\log\lambda} $ Riemann (1868, p. 89): $ \qquad \dfrac{d^2y}{dx^2}-\dfrac1{\alpha\alpha}\dfrac{d^2y}{dt^2}=...
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8 votes
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How and when was Bolzano's proof of the Bolzano-Weierstrass theorem rediscovered?

Fair warning: this answer doesn't completely answer the question, but I think it may answer the question as well as it is possible to do. The article which Hankel wrote (published 1971) that is ...
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8 votes

Is Kline right that Cauchy believed that continuous functions must be differentiable?

I do not believe Cauchy wrote anything explicit concerning the extent to which we can deduce differentiability from continuity. However, in many places it seems that Cauchy assumes continuity and then ...
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8 votes

What did Kurt Gödel mean by nonstandard analysis being "the analysis of the future"?

In the 1970s Gödel recommended Robinson for membership in the British Academy of Sciences. His recommendation was based in part on Robinson's framework for analysis with infinitesimals. From the ...
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  • 3,639
8 votes

What brought about the need for real analysis and formal logic in recent years?

As I explained in my answer to your other question, mathematics was always done using ordinary (non-formalized) logic. Attempts to formalize logic begin with Aristotle. (This is called "formal logic")....
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8 votes
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Did Cauchy ever deal with double or triple integrals?

Yes, he did, multiple times. Singular double integrals (1814) In Mémoire sur les intégrales définies (1814) Cauchy studied why switching the order of integration in a double integral can sometimes ...
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8 votes

Historically, what led to the question of the validity of interchange of limit operations?

G. H. Hardy wrote in A Course in Pure Mathematics in Appendix II, section A note on double limit problems Let us consider some special instances. In $\S\ 213$ we proved that \begin{align*} \log(1+x)=...
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7 votes

Why did mathematicians not see that $f_n(x)=x^n$ is a counterexample to Cauchy's "theorem" about limits of continuous functions?

Cauchy's formulation of the theorem in 1821 is ambiguous and at any rate in a 1853 article Cauchy seems to state that it was incorrect as stated. On the other hand, in the 1853 article Cauchy modifies ...
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  • 3,639
7 votes

Is Kline right that Cauchy believed that continuous functions must be differentiable?

Many mathematicians did believe that continuous functions are mostly differentiable (except for some isolated points perhaps) until Weierstrass made a splash by publishing an example of a nowhere ...
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  • 65.8k
7 votes

$\epsilon$-$\delta$ definition of continuity

Regarding the use of $\epsilon$ and $\delta$ in the context of the definition of continuity of functions, we can say that they "were in the air" since Cauchy. See Augustin-Louis Cauchy's ...
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7 votes
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Who was the first to prove the mean value theorem?

As per your comment requesting details of the linked paper : Besenyei's paper begins with a history of the development of Rolle's Theorem into its general form. It follows this with the history of ...
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  • 5,984
7 votes

What was Lebesgue's original definition of a measurable set?

See : Thomas Hawkins, Lebesgue's theory of integration, AMS (1975), page 122, regarding Lebesgue's thesis "Intégrale, longueur, aire," published in 1902: $m(E)$ is a nonnegative measure on bounded ...
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7 votes
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What was Lebesgue's original definition of a measurable set?

Lebesgue used what Caratheodory called the outer measure. Fermat's library has an annotated translation of Lebesgue's 1901 paper On a Generalization of the Definite Integral, he is much briefer than ...
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7 votes
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Were there proofs of the Lebesgue Differentiation Theorem without using maximal functions?

The original proof appears in Lebesgue's Lecons sur l'integration et la recherche des fonctions primitives, Paris, 1904, freely available if you read French. He only considered continuous monotone ...
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7 votes
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How did Peano prove his existence theorem without Ascoli's theorem?

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, ...
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