Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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21
votes
2answers
755 views

Lebesgue density strictly between 0 and 1

I am having trouble with the following problem: Let $A\subseteq \mathbb{R}$ be measurable, with $\mu(A)>0$ and $\mu(\mathbb{R}\backslash A)>0$. Then how do I show that there exists $x\in ...
20
votes
8answers
2k views

How I can prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2?

Prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2 My attempt I proved that the sequence is increasing and bounded by 2, can anyone help me show that the ...
20
votes
6answers
3k views

Prove that $\sqrt 2 +\sqrt 3$ is irrational. [duplicate]

Please prove that $\sqrt 2 + \sqrt 3$ is irrational. One of the proofs I've seen goes: If $\sqrt 2 +\sqrt 3$ is rational, then consider $(\sqrt 3 +\sqrt 2)(\sqrt 3 -\sqrt 2)=1$, which implies ...
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 $f(a)$ ...
20
votes
2answers
609 views

The series $\sum_{n=1}^{+\infty}\frac{1}{1^2+2^2+\cdots+n^2}.$

How to justify the convergence and calculate the sum of the series: $$\sum_{n=1}^{+\infty}\frac{1}{1^2+2^2+\cdots+n^2}.$$
20
votes
4answers
941 views

Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only

I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\in ...
20
votes
4answers
1k views

Is there a $\sigma$-algebra on $\mathbb{R}$ strictly between the Borel and Lebesgue algebras?

So, after proving that $\mathfrak{B}(\mathbb{R})\subset \mathfrak{L}(\mathbb{R})$, I asked myself, and now asking you, is there a set $\mathfrak{S}(\mathbb{R})$, which satisfies: ...
20
votes
5answers
1k views

Approximation to $ \sqrt{2}$

I'm a first year Undergraduate student from India. Our professor is going to start a Real Analysis course in September and I was preparing for the initials. I tried and solved many problems, but this ...
20
votes
2answers
1k views

Enumerations of the rationals with summable gaps $(q_i-q_{i-1})^2$

Here is a question from my undergraduate days which I never knew the answer to. I just want to know if anyone can offer me a hint. Consider the rationals in $[0,1]$. Does there exist a (bijective) ...
20
votes
3answers
2k views

$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$

Trying to solve $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges. I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$ Would ...
20
votes
2answers
490 views

Distance from $x^n$ to lesser polynomials

I am interested in the $L_1$ distance of $x^n$ to the $\mathbb R$-span of $\{1,x,\ldots,x^{n-1}\}$ over some interval. We can WLOG consider the interval $[0,1]$ (say) because scaling and shifting only ...
20
votes
4answers
10k views

max and min versus sup and inf

What is the difference between max, min and sup, inf?
20
votes
2answers
433 views

When does $(uv)'=u'v'?$ [duplicate]

In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this dream product rule true? There are ...
20
votes
1answer
403 views

How to find $\int_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx$

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx. $$ (I have literature on this, if people want). Note, we can ...
20
votes
4answers
713 views

How does this discontinuity occur in evaluating a nested square root?

This question is based on a comment I made on a question likely to be closed. Let $$y=\sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \dots}}}}}$$ be the classic nested square root which has ...
20
votes
1answer
707 views

Integral $\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}\mathrm dx $

Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}\mathrm dx $$ analytically. We can also write $$ ...
20
votes
4answers
351 views

Continuous $f\colon [0,1]\to \mathbb{R}$ all of whose nonempty fibers are countably infinite?

I have been told it is possible to construct a continuous function $f\colon [0,1]\to \mathbb{R}$ such that $f^{-1}(x)$ is either empty or has cardinality $\aleph_0$ for every $x\in \mathbb{R}$. I've ...
20
votes
4answers
524 views

Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$

Inspired by Ramanujan's problem and solution of $\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \ldots}}}$, I decided to attempt evaluating the infinite radical $$ \sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}} ...
20
votes
2answers
533 views

If $f:\mathbb{R}\to\mathbb{R}$ is a left continuous function can the set of discontinuous points of $f$ have positive Lebesgue measure?

If $f:\mathbb{R}\to\mathbb{R}$ is a left continuous function can the set of discontinuous points of $f$ have positive Lebesgue measure? I wondered this today, but made little progress. Thank you.
20
votes
1answer
607 views

real analysis function takes on each value twice?

I want to find a function $f:[0,1] \to [0,1]$ such that $f$ takes on each value in $[0,1]$ exactly twice. I think this means there are an infinite number of discontinuities. Can anyone help me figure ...
20
votes
1answer
276 views

A combination integral and series resulting the inverse tangent integral

$\def\Ti{{\rm{Ti}}_2}$I have been able to solve an integral problem, now I tried to use the other method to crack the integral and I have to prove the following expression \begin{equation} ...
19
votes
3answers
964 views

an inequality: $1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53$

$n$ is a positive integer, then $$1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53.$$ please don't refer to the famous $1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$. I want to find a ...
19
votes
4answers
1k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...
19
votes
7answers
3k views

Why do introductory real analysis courses teach bottom up?

A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and ...
19
votes
4answers
10k views

Discontinuous derivative.

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is ...
19
votes
5answers
2k views

Perfect set without rationals

Give an example of a perfect set in $\mathbb R^n$ that does not contain any of the rationals. (Or prove that it does not exist).
19
votes
5answers
491 views

Computing $\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$

What ways would you propose for the limit below? $$\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$$ Thanks in advance for your suggestions, hints! Sis.
19
votes
6answers
1k views

Proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$

Thomson et al. provide a proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$ in this book. It has to do with using an inequality that relies on the binomial theorem. I tried to do an alternate proof ...
19
votes
2answers
1k views

Is a differentiable function on $(-2, 4)$ always integrable on $[-2, 4]$?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
19
votes
6answers
1k views

$f\colon\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(f(x))=x^2$ for all $x$?

A friend came up with this problem, and we and a few others tried to solve it. It turned out to be really hard, so one of us asked his professor. I came with him, and it took me, him and the ...
19
votes
2answers
1k views

“Pseudo-Cauchy” sequences: are they also Cauchy?

I tried to prove something but I could not, I don't know if it's true or not, but I did not found a counterexample. Let $(a_n)$ be a sequence in a general metric space such that for any fixed $k ...
19
votes
6answers
7k views

What is the 'implicit function theorem'?

Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why ...
19
votes
4answers
2k views

Sine function dense in $[-1,1]$

We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by ...
19
votes
4answers
460 views

Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$

I have spent my holiday on Sunday to crack several integral & series problems and I am having trouble to prove the following integral \begin{equation} \int_0^1 ...
19
votes
2answers
1k views

Tough contest problem

I found this problem in a collection of contest problems of a Russian competition in 1995 and wasn't able to solve it. Solve for real $x$: $$ \cos (\cos (\cos (\cos(x))))=\sin (\sin (\sin (\sin ...
19
votes
2answers
1k views

Olympiad calculus problem

This problem is from a qualifying round in a Colombian math Olympiad, I thought some time about it but didn't make any progress. It is as follows. Given a continuous function $f : [0,1] \to ...
19
votes
6answers
1k views

A question on Terence Tao's representation of Peano Axioms

While reading Terence Tao's book on Analysis I had some questions regarding the implication of the Peano Axioms. After writing the following four axioms (which I will write without changing their ...
19
votes
5answers
503 views

$f^2+(1+f')^2\leq 1 \implies f=0$

Find all $f\in C^1(\mathbb R,\mathbb R)$ such that $f^2+(1+f')^2\leq 1$ It's quite likely the answer is $f=0$. Note that $|f|\leq 1$ and $-2\leq f'\leq 0$. Therefore $f$ is decreasing and ...
19
votes
4answers
895 views

thoughts about $f(f(x))=e^x$

I was thinking, inspired by mathlinks, precisely from this post, if there exists a continuous real function $f:\mathbb R\to\mathbb R$ such that $$f(f(x))=e^x.$$ However I have not still been able to ...
19
votes
2answers
881 views

mystery regarding power series of $\frac{1}{\sqrt{1+x^{x}}}$

In the course of playing around with $\sum_{n=1}^{\infty} \frac{1}{\sqrt{1+n^{n}}}$, I used w|α to obtain the power series for $f(x)=\frac{1}{\sqrt{1+x^{x}}}$ which is \begin{align*} ...
19
votes
2answers
7k views

Does convergence in $L^{p}$ implies convergence almost everywhere?

If I know $\|f_{n}(x) - f(x)\|_{L^{p}(\mathbb{R})} \rightarrow 0$ as $n \rightarrow \infty$, do I know $\lim_{n \rightarrow \infty}f_{n}(x) = f(x)$ for almost every $x$?
19
votes
4answers
940 views

Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx = \frac{\pi^3}{16}-3G\log 2 \tag{1}$$ where $G$ is Catalan's Constant. I was able to express it in terms of ...
19
votes
4answers
662 views

Limit of series involving ratio of two factorials

$$ \sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3} $$ The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
19
votes
2answers
2k views

Comparing the Lebesgue measure of an open set and its closure

Let $E$ be an open set in $[0,1]^n$ and $m$ be the Lebesgue measure. Is it possible that $m(E)\neq m(\bar{E})$, where $\bar{E}$ stands for the closure of $E$?
19
votes
4answers
335 views

An integral by O. Furdui $\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$

The following integral was proposed in a paper by O. Furdui, namely $$\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$$ and then the generalization $$\int_0^1 \log^2(\sqrt[k]{1+x}-\sqrt[k]{1-x}) \ ...
19
votes
2answers
610 views

Function $f(x)=\int_0^\infty\left|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\right|\,dt$

Let $$f(x)=\int_0^\infty\Big|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\Big|\,dt,$$ where $|\dots|$ denotes the absolute value. We are concerned only with positive values of $x$ (i.e. let the domain of the ...
19
votes
1answer
495 views

Is there a function having a limit at every point while being nowhere continuous?

Is there a function $\,f:\mathbb{R}\rightarrow\mathbb{R},\,$ that has a limit at every point but is continuous nowhere?
19
votes
1answer
510 views

Functions $f$ satisfying $ f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R}$.

How to prove that the continuous functions $f$ on $\mathbb{R}$ satisfying $$f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R},$$ are given by $$f(x)=x+a,a\in\mathbb{R}.$$ Any hints are welcome. Thanks.
19
votes
2answers
1k views

Two definitions of Lebesgue integration

Normally, Lebesgue integral, for positive measures, is defined in the following way. First, one defines the integral for indicator functions, and linearly extend to simple functions. Then, for a ...
19
votes
2answers
477 views

Inequality in a bounded real sequence

Prove or disprove that for any bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ there exist two distinct natural numbers $u,v$ such that: $$|x_u-x_v|\cdot|u-v|\leq 1.$$