0
votes
0answers
24 views

List of crucial results deserving more attention for first course in Real Analysis

Can it help to form a list of crucial results for basic courses that are concealed as exercises or neglected? I don't know of other resources for this, as I wrote here. I am happy for this to be ...
2
votes
0answers
84 views

List of Common or Useful Limits of Sequences and Series

There are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required. ...
1
vote
0answers
149 views

Lecture Notes in Real Analysis

I understand that this question was partially addressed here but I would like to have a question dedicated to just real analysis. I am looking for both elementary real analysis (advanced calculus type ...
1
vote
0answers
28 views

limit formalisms

Let $f:\mathbb R \to \mathbb R$ be a function and $a\in \mathbb R$ a point. The Cauchy definition of the limit $\lim _{x\to a}f(x)=L$ is well-known. For pedagogical reasons I'm interesting in a ...
3
votes
1answer
205 views

Open Problems in Real Analysis [closed]

What are some open problems in Real Analysis? I have found some on the Open Problem Garden, but would like to see some more.
0
votes
0answers
107 views

Find two linearly independent functions with a zero Wronskian but a nonzero product.

Known to me examples of L.I. functions having Wronskian=0 have also product=0. One such example was manufactured by Peano about 1890. By the way: Analytic functions with a zero Wronskian are linearly ...
18
votes
7answers
4k views

Any open subset of $\Bbb R$ is a countable union of disjoint open intervals. [Collecting Proofs]

This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...
20
votes
5answers
1k views

Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis

I am studying for an exam and I have been studying my butt off during the winter break for it. During the course of my study I have written down quite a number of tricks, which in my opinion were ...
3
votes
1answer
168 views

Nice applications of the Haar measure

The existence of the Haar measure is a beautiful result that has a lot of applications. For example, one can prove using the Haar measure that the category of representations of a compact Lie group is ...
1
vote
0answers
66 views

Partition of open sets in $\mathbb{R}^d$.

Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets. In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
2
votes
1answer
81 views

Inequalities involving some common functions

I often see the following inequality is used over and over again $$ 1−x⩽e^{−x} $$ for $x \in \mathbb{R}$, for proving or deriving various statements. As a layman, I haven't seen this inequality ...
21
votes
1answer
693 views

Expository articles on Analysis and Probability theory

When I come across a notion from algebra or number theory which I don't know I usually check Keith Conrad's page to see if he has written something about it. Key features of his articles are a very ...
14
votes
2answers
485 views

open conjectures in real analysis targeting real valued functions of a single real variable

I am hoping that this question (if in acceptable form) be community wiki. Are there any open conjectures in real analysis primarily targeting real valued functions of a single real variable ? (it may ...
0
votes
1answer
409 views

Self-Contained Treatments of Stokes's Theorem for Manifolds [closed]

I am seeking to compile a list of textbooks that provide self-contained treatments of Stokes's Theorem in the language of differential forms and manifolds. By "self-contained", I mean the statements ...
8
votes
2answers
812 views

Proofs of the Cauchy-Schwarz Inequality?

How many proofs of the Cauchy-Schwarz inequality are there? Is there some kind of reference that lists all of these proofs?
20
votes
15answers
3k views

Useful examples of pathological functions

What are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples? For instance, if $f'(a) = b$, then $f(a)$ ...