# 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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### Was there a more intuitive early proof of the generalized mean value theorem?

I am interested in the early proofs of the theorem. It is often called Cauchy mean value theorem, so perhaps Cauchy proved it first. In all the proofs that I have seen we construct a contrived ...
1answer
107 views

### How did Peano prove his existence theorem without Ascoli's theorem?

In modern proofs of the Peano Existence Theorem for ordinary differential equations, Ascoli's theorem is used. Ascoli's theorem came after Peano's proof. Did Peano prove a form of Ascoli's theorem in ...
0answers
87 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 ...
2answers
148 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 ...
2answers
164 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?
1answer
75 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 ...
1answer
107 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 ...
3answers
316 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 ...
3answers
133 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, ...
2answers
93 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 ...
1answer
162 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 ...
1answer
50 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?...
2answers
429 views

### Dirichlet's Proof of the Convergence of Fourier Series

Where can I find Dirichlet's proof of the convergence of Fourier series?
1answer
63 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 ...
2answers
156 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 ...
1answer
120 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?
2answers
285 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 ...
1answer
78 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 ...
2answers
442 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 ...
1answer
120 views

### The Origin of the Jacobian

In what work did Jacobi introduce the jacobian, and what was his motivation for doing so?
0answers
77 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 ...
1answer
73 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.
1answer
187 views

### Motivation of Continuous Functions

What is the historical motivation of continuous functions? Also, does anyone know who first isolated the idea?
1answer
238 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}\...
2answers
333 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 ...
1answer
188 views