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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Continuity Must Hold in an Entire Open Set?

Claim: If a function $\mathbb{R^n} \rightarrow \mathbb{R^m}$ is continuous at $\vec a \in \mathbb{R^n}$, it is continuous in some open ball around $\vec a$. Is this claim correct? In other words, is ...
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0answers
31 views

Proving that a sequence converges or diverges

Prove or disprove that there is a sequence $n_k$ of positive integers such that both $\cos(2n_k!)$ and $\sin(2n_k!)$ converge. I think that the series diverges but I am not sure how to prove it.
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1answer
25 views

Approximate function as $x$ tends to infinity

I'm looking for a way to approximate the following function $f$ as $x \to \infty$ $$ f = \ln \left( 1 + e^{a_1 x} + e^{a_2 x} + A e^{(a_1+a_2) x} \right) $$ where $a_1$, $a_2$ and $A$ are constants. ...
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0answers
8 views

Proving the last part of Nested interval property implying Axiom of completeness

I took a non-empty set A that is bounded above. And I went on with the regular algorithm, which either gave us a LUB or gave us an infinite chain of nested intervals $I_1$ $\supseteq$ $I_2$ ...
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1answer
27 views

if the integrals of a non-negative sequence of functions go to zero, does this imply functions go to zero a.e.? [duplicate]

This question arised when I was dealing with an old qual problem, and if this is true, I'll be done, but I'm not sure if it's true or not: Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of ...
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0answers
18 views

Necessary to assume $f\in C^\infty$ in this Fourier transform problem?

Consider the following problem. Is the hypothesis that $f\in C^\infty$ necessary, or could we weaken it and assume just that $f$ is continuous? Let $\hat f$ denote the Fourier transform of the ...
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1answer
49 views

Doesn't $x^3+2y^3+3z^3=0$ give a surface in $R^3$?

In my last exam on Advanced Calculus (following Spivak's Calculus on Manifolds), I couldn't solve the following question. True or false: the set $S$ in $R^3$ given by $x^3+2y^3+3z^3=0$ is a ...
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1answer
100 views

What star domain has a non-star-domain interior?

Definition: We call a subset $S$ of $\mathbb{R}^n$ a star domain (or star-shaped) if there exists a point $x_0 \in S$ such that for every $x \in S$, the line segment $\overline{x_0x}$ is contained ...
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1answer
32 views

$ |x_{n} - y_{n}| < \frac{1}{n} \Rightarrow |x'_{n} - y'_{n}| < \frac{1}{n}$

This came from a proof on uniform continuity theorem. My textbook claims that if sequence $(x_j)$ and $(y_j)$ on a compact set D have the condition that $ |x_n - y_n| < \frac{1}{n}$, and $(x'_j)$, ...
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1answer
24 views

Dense Domain: Preimage

Given Banach spaces $X$ and $Y$. Regard a bounded operator: $$A\in\mathcal{B}(X,Y)\implies A\in\mathcal{C}(X,Y)$$ Then for dense sets: $$W\leq Y:\quad \overline{W}=Y\implies\overline{A^{-1}W}=X$$ ...
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147 views

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...
2
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3answers
54 views

Multiplication operator on $L^1$

Let $\phi :X \rightarrow \mathbb{C}$ be measurable with respect to the measure space $(X,\mu)$. Suppose that $\phi f \in L^1(\mu)$ whenever $f \in L^1(\mu)$. Define $M_{\phi}(f)=\phi f$, for $f \in ...
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0answers
23 views

$\liminf_n a_n = \inf_n a_n$ if $a_n \ge a_m$ when $n\mid m$

I would like to ask a reference for the following very easy result: can someone help? Let $(a_n)_{n\ge 1}$ be a sequence of positive reals such that $a_m \le a_n$ whenever $n$ divides $m$. Then $$ ...
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1answer
21 views

Bounded harmonic function on $\mathbb{R}^3$

Any suggestions how to get started? I know Liouville's theorem, but not sure how to apply it here: Let $u$ be a harmonic function on $\mathbb{R}^3$. Assume there exists $C>0$, independent of $x$, ...
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3answers
52 views

Showing that a functions derivative is not bounded on $\mathbb{R}$

Suppose that $f$ is differentiable but not uniformly continuous on $\mathbb{R}$. Prove that $|f'|$ is not bounded on $\mathbb{R}$. So I know that to show that $|f'|$ isn't bounded you would have to ...
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1answer
37 views

Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \le 0$. Proof: I will attempt to show that the ...
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0answers
24 views

Compute the limit (using LDCT) [duplicate]

Could you please help me with the following? I think it is a Lebesgue Dominated Convergence Theorem but I am not sure. Compute the following: For $1\leq p<\infty$ and $f\in L^p(\mathbb{R})$ ...
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2answers
79 views

Prove that Euclidean distance in $\mathbb{R}^n$ is a distance

I'm trying to show that: $$\forall x,y\in\mathbb{R}^n, d(x,y)=\left(\sum_{i=1}^n(x_i-y_i)^2\right)^{1/2}$$ is a distance. However I have not proved Cauchy-Schwarz yet and I'm pretty sure I wouldn't ...
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4answers
77 views

Show that $(1+p/n)^n$ is a Cauchy sequence for arbitrary $p$

It is a generalization of this question. I am looking for a similar derivation as in here. Can we prove that $(1+p/n)^n$ is a Cauchy sequence for any $p \in [a, b]$ by showing that $$ \Bigg| \left( ...
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0answers
26 views

Show that there exists a $g \in L^1(m)$ s.t. $\phi (F(x))=\int_{0}^{x} g(t)dt$, where $F(x)=\int_{0}^{x} f(t)dt$

Let $m$ be Lebesgue measure on $[0,1]$ and suppose $f \in L^1(m)$ and let $F(x)=\int_{0}^{x} f(t)dt$. Suppose $\phi$ is a Lipschitz function. Show that there exists a $g \in L^1(m)$ s.t. $\phi ...
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17 views

The dual function of composite functions

Given $X$ $Y$ are two finite dimensional Hilbert space. Let $K$: $X\to Y$ be linear and $F$: $Y\to \mathbb R^+$ is convex. Let us use $F^\ast$ to denote the dual (conjugate) function of $F$. Recall $$ ...
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2answers
49 views

Euclidean Spaces: Embedding

Given the real line $\mathbb{R}$ and plane $\mathbb{R}^2$. Are there maps: $$\eta\in\mathcal{C}(\mathbb{R}^2,\mathbb{R}),\vartheta\in\mathcal{C}(\mathbb{R},\mathbb{R}^2):\quad ...
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2answers
12 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
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1answer
32 views

Problem in showing that a sequence is a Cauchy sequence on a space with the integral metric.

I'm having difficulty following what is going on and understanding parts in the following example. It is quite similar to an example I posted before (Changing of the limits of integration with the ...
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0answers
23 views

Reverse Intermediate Value Theorem

What does it mean to say that a real valued function $ f : [a, b] \rightarrow \mathbb{R} $ is continuous at $ x_0 \in [a, b] $? Assume that $ f : [a, b] \rightarrow \mathbb{R} $ is continuous State, ...
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24 views

The relationship between outer measures and smallest coverings

Recall that if $\mathcal{A}\subset \mathcal{P}(X)$ is an algebra and $\mu_{0}:\mathcal{A} \to [0,\infty]$ is a premeasure on $\mathcal{A}$ then we can define the outer measure $\mu^{*}$ for any set ...
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1answer
14 views

Preference relations and the existence of extensions of functions representing them

In a book I found the following question: Let $\succsim$ be a complete preference relation on a nonempty set $X$, and let $\varnothing \neq B \subseteq A \subseteq X$. If $u \in [0,1]^A$ ...
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1answer
27 views

Help with a problem regarding sequence divergence.

There are two forms of definition of sequence divergence. By negation of the sequence convergence we have A sequence $x_k$ diverges iff $∀x∈\Bbb{R}∃ϵ>0∀N∈\Bbb{N}∃k>N$ st. $|x_k-x|>ϵ$. ...
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1answer
41 views

Understanding Cohn's Radon-Nikodym proof from his book on measure theory

The part of the proof which I don't get is $$\nu(A)=\int_{A} g\ \mathsf d\mu$$ where $g$ is Radon-Nikodym derivative. He has a set of functions for which $$\int_{A} f\ \mathsf dx \le \nu(A) ,$$ he has ...
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1answer
36 views

Intuition behind surface integrals

While line integrals derive their intuition from , and are analogous to, the concept of Work in physics, what intuition is there to back up the notion of surface integrals? In the texts I've been ...
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0answers
36 views

Discontinuities of an injective funtion from $\mathbb{R}$ to $\mathbb{R}$

It is well known that a monotonic function from $\mathbb{R}$ to $\mathbb{R}$ can have only countably many discontinuities. Question: Is it true that an injective function from $\mathbb{R}$ to ...
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0answers
18 views

Derivative and monotonicity [on hold]

continuity in [a,b] and non negativity of one of Dini derivative in (a,b) implies the function is nondecreasing on [a,b].This is a statement in Royden,but I am not able to prove it,Plz help me
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1answer
32 views

Is there any way to combine the product of two univariate integrals into a single integral?

Can two separate integrals but multiplied together in the end by integrated as a product once instead? In other words, does $$ \left(\int_{-\infty}^{+\infty}f(x)\mathrm{d}x ...
4
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1answer
61 views

Specific problem on distribution theory.

*****Note: Parts A, C and D I managed. Only need help on part B now would really would appreciate the help on B Hi, in my summer real analysis (or measures and real analysis as my instructor refers ...
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2answers
22 views

Convegence of $\sum_{i\in J}a_i$ implies that index set is countable

Let $J$ be a uncountable set and $\{a_i\}_{i\in J}$ be a set of non-negative real numbers. Prove that $\sum_{i\in J}a_i<\infty$ implies that there is a countable set $H\subset J$ such that $a_i=0$ ...
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1answer
23 views

Under the Borel measure associated to the Cantor function each of the intervals remaining in the construction of the Cantor set has measure $2 ^{-n}$

Let $f$ be a function such that agrees with the cantor function on $[0,1]$, vanishes on $(-\infty,0)$, and is identically $1$ on $(1,+\infty)$ and let $\mu_f$ the Borel measure associated to $f$. Show ...
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1answer
20 views

Question on mutual singularity and absolute continuity of complex measures

I was presented these two somewhat similar questions from Folland's real analysis (second edition) dealing with complex measures and their mutual singularity and absolute continuity. They are 3.19 and ...
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3answers
79 views

Challenging $\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}$ for $\epsilon=\frac{1}{2}$.

Consider the (incorrect) claim that $$\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}.$$ How might I find the largest $\delta$ such that I can challenge $\epsilon = 1/2$? Clearly ...
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1answer
21 views

Convexity of $ t \mapsto \log \left[ \int_0^1 f^{pt} g^{q(1-t)} dx \right] $

Let $f,g \geq 0$ be bounded, measurable functions on $[0,1].$ For real $p,q>0$ with $p^{-1}+q^{-1} = 1$, I want to show the convexity of $$ h(t) = \log \left[ \int_0^1 f^{pt} g^{q(1-t)} dx ...
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3answers
83 views

Computing $\sum_{n=-\infty}^\infty \int_{-\pi}^\pi e^{-x^2} \cos(nx) dx $

I'm trying to find $$ \sum_{n=-\infty}^\infty \int_{-\pi}^\pi e^{-x^2} \cos(nx) dx. $$ About the only thing I can think of is the well-known identity $$ \sum_{n=-k}^k \cos(nx) = \sum_{n=-k}^k ...
5
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2answers
56 views

A problem related to integration in $L^1$

If $f\in L^1[0, 1]$ and $\int_{0}^1 x^nf(x)=0$ for all $n = 0,1,2,...$then prove that $f$ is identically zero almost everywhere. This would be very easier to prove if $f$ were continuous on $[0, 1]$ ...
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1answer
50 views

Application of Baire Category theorem

Suppose that $f$ is infinitely differentiable on $[a,b]$ and suppose that for any $a ≤ x ≤ b$ the Taylor series of $f$ has positive radius of convergence at $x$. Use the Baire Category Theorem to ...
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3answers
196 views

Not the toughest integral, not the easiest one

Perhaps it's not amongst the toughest integrals, but it's interesting to try to find an elegant approach for the integral $$I_1=\int_0^1 \frac{\log (x)}{\sqrt{x (x+1)}} \, dx$$ $$=4 ...
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0answers
68 views

Can $\int_{0}^{1}\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx$ be expressed in a simple form?

I was browsing the book Irresistible Integrals and found this gem, at page 97, $$ \int_{0}^{1}x^{n}\ln^{k}(x)dx=\frac{(-1)^{k}k!}{(n+1)^{k+1}} $$ that resembles a previous question of mine here. So, ...
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0answers
30 views

Example of a well defined functional integral?

So I was playing around with the notion of a functional integral. Basically given a set $S$ of functions we can define $$ \int_{f \in S} L(f) $$ As the sum of of every function $f$ evaluated by ...
2
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1answer
53 views

Find $ \lim_{x\to 0^+ } \frac{\alpha(x)}{x}$

Let $f$ be a continuously differentiable function with $f'(0)\neq 0$. Let $x>0$ and define $$ \int_{0}^x f(u) du = f(\alpha) x $$ where $\alpha=\alpha(x)$ is a number in $[0,x]$. Find $$ ...
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2answers
63 views

A question about bijective functions

Suppose that $f: [1,3]$$\rightarrow$$[0,8]$ is continuous. Show that there is some $x$$\in$$[1,3]$ such that $f(x)+4=4x$. I need some advice on how to get started. Do i need to use IVT for this ...
0
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0answers
32 views

Help verify a proof about “extension of monotone function”

For any monotone function $g$ defined on a non-empty set $\Omega\subset\Bbb{R}$, show that there exists a monotone function $f$ defined on $\Bbb{R}$ such that $f=g$ on $\Omega$. The monotonicity of ...
4
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1answer
55 views

Differentiability at a point $x$ with $f$ differentiable in $\mathbb R\backslash\{x\}$

I have a real function that satisfies: $f:\mathbb R\rightarrow\mathbb R$ is differentiable at $x$ for $x\neq x_0$. There is a full-measure set $T$ such that for any sequence $t_n\in T$ with ...
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1answer
20 views

Property of an almost additive sequence of functions

We say that a sequene of functions $\Phi=(\phi_n)_n$ is almost additive if there exists a constant $C > 0$ such that for every $n,m \in \mathbb{N}$ and $x\in \Lambda$ we have \begin{equation*} -C + ...