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Questions tagged [real-analysis]

For questions about the history of calculus and its theoretical foundations, including topics such as continuity, differentiability, and infinite series. Related topics include questions on the history of measure theory, and some aspects of general topology and classical descriptive set theory.

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41 views

The Integral as a Uniform Limit of Step Functions

Who first realized that it is possible to define the integral of a function as the limit of the integrals of a sequence of step functions that converge uniformly to the given function? This is ...
2
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2answers
70 views

Riemann's Contribution to Integration

What did Riemann do for the theory of integration? I am asking because I hear his name a lot in relation to integration and it is often implied that he made large contributions, but I do not know ...
5
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2answers
99 views

How did Newton and Leibniz interpret the integral?

How did Newton and Leibniz think about the integral? Did they only see it as an anti-derivative or did they also think of it as the area under a curve?
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1answer
64 views

What geometric results were first proven by assuming all real numbers are rational?

Pythagoras and his followers believed that all magnitudes are commensurable; that is, the ratio of two magnitudes of the same kind, like two lengths or two areas, is equal to the ratio of natural ...
2
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1answer
66 views

Who first proved Fubini's theorem for abstract measure spaces?

Fubini's theorem relates the double integral of a function $f(x,y)$ to an iterated integral with respect to $x$ and $y$. The basic idea of this theorem for Riemann integrals of continuous functions ...
4
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3answers
283 views

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

I can't seem to find a clear, definitive, non-circular answer on this. For centuries and centuries, we've been doing mathematics in one form or another, be it geometry and pictures, or inventing ...
3
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3answers
89 views

Did Eudoxus really set out to partition irrationals (Dedekind cuts) with rationals or was that a mere side effect we perceive through our modern POV?

I've been intrigued by the similarities between what Eudoxus' Theory of Proportions and Dedekind cuts. However, I wish to question this "perceived similarity" and would like to where the flaws are, ...
5
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2answers
82 views

Does anybody know the history of how Peter Gustav Lejeune Dirichlet came up with the “nowhere continuous” Dirichlet function?

So I am writing a research paper on the properties of the Dirichlet function (the function with 1 if x is rational and 0 if x is irrational), and I wanted to include some historical background on how ...
2
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1answer
94 views

An english translation of Cauchy's “Cours d'Analyse”

I am quite interested in the origins of our modern way of understanding analysis. I know that Augustin-Louis Cauchy was one of pioneers regarding a rigorous foundation towards real and complex ...
3
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1answer
40 views

Were there proofs of the Lebesgue Differentiation Theorem without using maximal functions?

Is there a proof of the Lebesgue Differentiation Theorem that does not involve the Hardy-Littlewood Maximal Function? For example, did Lebesgue prove it? If there is such a proof, where can I find it?...
2
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2answers
193 views

Dirichlet's Proof of the Convergence of Fourier Series

Where can I find Dirichlet's proof of the convergence of Fourier series?
2
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1answer
49 views

Reference - Schwarz's Proof of Clairaut's Theorem

Where can I find a copy (online) of Schwarz's paper that proved Clairaut's theorem for mixed partial derivatives? His paper is: Schwarz, H. A., "Communication", Archives des Sciences Physiques et ...
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2answers
113 views

What is history behind Smith-Volterra-Cantor sets?

Looking at Wikipedia, I see that fat Cantor sets are also called Smith-Volterra-Cantor sets. Another name which is sometimes associated with these sets is Hermann Hankel. I suppose that Cantor's name ...
4
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1answer
81 views

Who proved the monotone convergence theorem for the Lebesgue integral?

The theorem often be called Lebesgue's MCT or Levi's theorem. Who did originally prove it or what is the contribution of Lebesgue and Levi respectively?
8
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2answers
190 views

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

There are several equivalent ways to define a measurable subset of $\mathbb{R}$. One way is to start with the Lebesgue outer measure and then restrict it to the set of subsets satisfying the ...
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1answer
62 views

What was Lipschitz's original motivation for the introduction of Lipschitz continuity?

The concept of Lipschitz continuous mappings is probably at the present time the most important mathematical concept associated with Lipschitz's name. These mappings play an important role in the ...
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2answers
312 views

What is the correct statement of Cauchy’s erroneous theorem on continuity?

I read recently that Cours includes a famous, or perhaps infamous, error in that Cauchy states and proves a false result concerning sequences of continuous functions. (Here, obviously, continuous ...
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1answer
92 views

The Origin of the Jacobian

In what work did Jacobi introduce the jacobian, and what was his motivation for doing so?
4
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0answers
75 views

Variants in graphical presentation of real and complex numbers

It's standard nowadays for the real line to be horizontal (negative numbers on the left, positive numbers on the right) and for $i$ to be above (rather than below) 0 in the complex plane. Were these ...
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1answer
65 views

Motivation of Infinite Series

What is the historical motivation of infinite series? According to Wikipedia, they are arose separately by Newton, Leibniz and Somayaji.
4
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1answer
185 views

Motivation of Continuous Functions

What is the historical motivation of continuous functions? Also, does anyone know who first isolated the idea?
7
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1answer
228 views

Historical occurrences of mathematicians substituting terms for $x$ in the denominator of $\mathrm{d}y/\mathrm{d}x$?

This answer, to a question on teaching the chain rule, suggests writing something like this $$ \frac{\mathrm{d}\, \mathrm{e}^\sqrt{s}}{\mathrm{d}\,s}=\frac{\mathrm{d} \,\mathrm{e}^\sqrt{s}}{\mathrm{d}\...
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2answers
264 views

Basic Theorems in Topology: Who proved them first?

Little thinking into basic Real Analysis results like Arbitrary union of open sets is open made me wonder who could have possibly proved it first - do we have any historic data on it? Also, who ...
4
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0answers
123 views

First appearance of Hadamard's lemma on smooth functions

Hadamard's lemma, in one dimension, says for any smooth function $f \colon \mathbf R \rightarrow \mathbf R$ there is a first-order expansion of $f$ at $0$: $f(x) = f(0) + xg(x)$ where $g \colon \...
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1answer
86 views

How did Dedekind arrive at the completeness of real numbers?

Reading Dedekind original manuscript (https://archive.org/details/essaysintheoryof00dedeuoft) it seems that he was after formalizing the completeness of the real numbers, which was known to him as the ...
4
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2answers
246 views

What did the ratio of two magnitudes mean to ancient Greek mathematicians?

My understanding is that magnitudes to ancient Greeks meant the actual line segments and plane regions (not the size of the line segment or the area of the plane region), the concept of ratio was then ...
7
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1answer
247 views

Who first proved the “Cantor-Heine theorem” on uniform continuity?

The theorem is that any continuous function on a compact is uniformly continuous. It is called "Heine", and sometimes also "Heine-Cantor" theorem. My question is: what is the contribution of Cantor ...
8
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1answer
219 views

What were the earliest “unpleasant” consequences of the Axiom of Choice (and its negation) to be deduced?

I read that Zermelo formulated AC in 1904 in order to formally prove the well-ordering theorem. Vitali’s 1905 proof of the existence of a non-measurable set of real numbers appears to the first “...
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1answer
102 views

Straight line is the shortest of curves, who proved?

I am curious, when and by whom it was proved that straight line is the shortest of measurable curves connecting two given points.
3
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1answer
971 views

What is the origin of the term “ordinary differential equations”?

Who has first used the term "ordinary differential equation"? Is it known, why the term "ordinary" is used here? What makes an ODE "ordinary"?
3
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1answer
129 views

First evaluation of $\sum_{n \geq 1} 1/n^2$ by Fourier series

There are many ways to evaluate $\sum_{n \geq 1} 1/n^2$ as $\pi^2/6$, including multiple solutions using Fourier series. A colleague asked me who was the first person to use Fourier series (or Fourier ...
11
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2answers
685 views

Who was the first to prove the mean value theorem?

Who was the first to prove the mean value theorem, i.e., the existence of an intermediate point where the slope equals the average slope over the interval?
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0answers
96 views

Felix Klein and the mean value theorem

This is a reference request prompted by some intriguing comments made by Felix Klein. In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be ...
0
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1answer
111 views

mathematics was recreated on a foundation of number concepts rather than geometrical ones

In Richard Courant and Fritz John's book Introduction to Calculus and Analysis Volume I, says In modern times mathematics was recreated and vastly expanded on a foundation of number ...
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5answers
1k views

Who gets credit for the real numbers?

If Simon Stevin already pioneered the unending decimal representation for every number (rational, surd, etc.) at the end of the 16th century, why do Cantor and Dedekind (who certainly gave a more ...
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2answers
382 views

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

I found this : "There are good reasons to believe that nonstandard analysis, in some version or other, will be the analysis of the future." What did Kurt Gödel mean exactly ?
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3answers
720 views

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

From time to time one sees insistence that the inequality name "Cauchy-Schwarz" should include Buniakovsky. This is based on a paragraph in a note to the St Petersburg Academy from 1859, where ...
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4answers
875 views

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

Morris Kline, in Mathematical Thought from Ancient to Modern Time, writes in chapter 40 (The Installation of Rigor in Analysis), "Though Bolzano and Cauchy had rigorized (somewhat) the notions of ...
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4answers
2k views

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

EDIT: (130116) I don't mean it is complex or imaginary nor it is negative also, I tried hard to conceive it on the real line number (positive X-axis), by obvious means, a little idea came to me?, "...
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2answers
394 views

$\epsilon-\delta$ definition of continuity

According to Wikipedia Bolzano and later Weierstrass were the first who gave an $\epsilon-\delta$ definition of continuity and convergence. But did they already use the letters $\epsilon$ and $\delta$ ...
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1answer
149 views

Where did d'Alembert published the ratio test?

The Wikipedia article ratio test states that it was first published by Jean le Rond d'Alembert. In which of his works did he state the ratio test?
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0answers
101 views

Are there any functions which were proposed as elementary by mathematicians but not considered elementary now?

Are there any functions which were proposed by various mathematicians to be included in the set of elementary functions because of their properties but not considered elementary as of now?
2
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1answer
455 views

Origin of Cauchy convergence test

Obviously Cauchy's convergence test is named after Augustin-Louis Cauchy. Is he the person who first proved this criterion or is it another misnamed theorem? If so: In which treatise?
2
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1answer
451 views

Where is the root test first proved

Concerning the Wikipedia article "root test", this convergence criterion for series was first proved by Augustin-Louis Cauchy. What is the name of the treatise where he proved the root test? Which ...
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0answers
303 views

When did people know that all real polynomials of degree greater than 2 are reducible?

Admittedly, this may not be a research level question, but I am deeply curious about this. Let $f(x) \in \mathbb{R}[x]$, and write $d = \deg f$. It is well known that if $\deg f > 2$, then $f$ is ...
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0answers
161 views

Who first used Banach's Contraction Principle to prove the Inverse Function Theorem?

The most common proof of the Inverse Function Theorem in textbooks relies on the Banach fixed-point theorem. QUESTION. It is possible to say, using historical references, which mathematician ...
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4answers
2k views

What was the answer to this paradox before Cantor?

I do not remember the name/source of this paradox,but I remember I have discussed this with mathematicians and non mathematicians at least 5 times. It goes like this: "Every point of a line has ...
3
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2answers
311 views

Who discovered the difference between the infinities?

As we know, there is a difference between the (infinite) size (or cardinality) of the integer numbers and the size of the reals ($\aleph_0$ and $\mathfrak c=2^{\aleph_0}$). Who discovered it first?
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3answers
616 views

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

In 1821 Cauchy claimed that the limit of a sequence of continuous functions is continuous. In 1826 Abel gave a complicated trigonometric counterexample. When we teach analysis courses, we usually give ...
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1answer
939 views

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

Some functions of a real variable are infinitely smooth (have derivatives of all orders) but are not analytic (at some points $a$, the Taylor series at $a$ does not represent the function at any ...