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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Question about a real monotone function extracted from the book “Curso de Análise I” by: Elon Lages Lima.

If $ f : X \subset \mathbb{R} \longrightarrow \mathbb{R} $ is a monotone function then the set of all points $ a \in X^{'} $ where ($ X^{'}$ is the set of all accumulation points of $X$) such ...
3
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1answer
39 views

Is the following integral identity true or not? [on hold]

Is the following statement true or not?$$\int_{-\infty}^\infty xf(x)\,dx = \left. {d\over{dt}} \int_{-\infty}^\infty e^{tx}f(x)\,dx\right|_{t = 0}$$
3
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2answers
52 views

Prove that a function is contractive

I'm stuck with the following. I need to prove that in $D:=[0,1]\times[0,1]$ the function $F$ is contractive, where $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is defined as: \begin{align} F(x,y):=(\frac{...
4
votes
2answers
51 views

Real Analysis, 2.18 (Fatou's Lemma) Integration of Nonnegative functions

2.18 Fatou's Lemma - If $\{f_n\}$ is any sequence in $L^+$, then $$\int \left(\lim_{n\rightarrow \infty}\inf f_n\right) \leq \lim_{n\rightarrow \infty}\inf\int f_n$$ Attempted proof - We know that $$...
3
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3answers
53 views

MVT for integrals: strict inequality not needed before applying IVT?

I've looked at Nigel Overmars's answer here: http://math.stackexchange.com/a/630429/349828 His proof is essentially identical to the one I wrote myself and to the one given by my analysis professor ...
2
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3answers
44 views

Understanding notation for the sequence definition

Looking for assistance in translating this definition into more laymen terms? In other words, can someone explain it to me like I'm a 5 year old? Definition. A sequence ($s_n$) is said to diverge ...
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1answer
31 views

Real Analysis, Folland Proposition 2.16 and Corollary 2.17 Integration of Nonnegative functions

Background Information: Proposition 2.16 - If $f\in L^+$, then $\int f = 0$ iff $f = 0$ a.e. Proof - Suppose $f = \sum_{j}a_j\chi_{E_j}$, then $\int f = 0$ iff $a_j = 0$ or $\mu(E_j) = 0$. In ...
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1answer
40 views

Real Analysis, Folland Theorem 2.14 (Monotone Convergence Theorem)

Theorem 2.14 (MCT) - If $\{f_n\}$ is a sequence in $L^+$ such that $f_{n}\leq f_{n+1}$ for all $n$, and $f = \lim_{n\rightarrow \infty}f_n (=\sup_n f_n)$, then $\int f = \lim_{n\rightarrow \infty}\int ...
3
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1answer
50 views

function is not differentiable on $\mathbb R\setminus\{0\}$

I need to prove that the given function $f$ is not differentiable on $\mathbb R \setminus\{0\}$. $$ f(x) = \begin{cases} x^2, \ x \in \mathbb{Q}\\ 0, \ x \in \mathbb{R}-\mathbb{Q} \end{cases} $$ ...
3
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1answer
75 views

Tricky norm-inequality $\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$ for $p \in (0,1)$

For $r>p \ge 1$ one can show that in $\mathbb{C}^n$ we have $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ My question is now: Does this also hold for $1 \ge r>p>0$? Obviously we ...
0
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2answers
70 views

Is f(x,y)=$\frac{x^{2}y}{x^{2}+y^{4}} $with f(0,0)=0 continuous in (0,0) [duplicate]

I believe that the function: f(x,y)=$\frac{x^{2}y}{x^{2}+y^{4}}$ is continuous on the point (0,0) but i can't prove it. I know you have to choose something like $x=cy^{2}$(with c a constant) to prove ...
0
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0answers
20 views

Distance between zeros of continuous function

Dears, Let $f(x):=1+\sum_{k=1}^{n}a_{i}b_{i}^{k}$, for each $x\in[a,b]$, where $a_{i}$ are real numbers (not nulls) and $b_{i}>0$. Assume that $f$ has, at least, two ceros in $[a,b]$. Then, I want ...
0
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1answer
22 views

a problem concerning continuous functions of bounded variation [duplicate]

Here is a problem: Suppose $f,g: [a,b]\rightarrow \mathbb{R}$ are both continuous and of bounded variation. Show that the set $\{(f(t),g(t))\in\mathbb{R}^2: t\in [a,b]\}$ CANNOT cover the entire unit ...
2
votes
2answers
43 views

Find $\lim_{(x,y)\to(0,0)} g \left(\frac{x^4 + y^4}{x^2 + y^2}\right)$ where $\lim_{z\to 0}\frac{g(z)}{z}=2.$

This limit seems different to me than all the other multi variable limits already asked on this site. Let $g \colon \mathbb R \to \mathbb R $ be such that $$ \lim_{z\to 0}\frac{g(z)}{z}=2. $$ ...
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1answer
44 views

Function measurable iff the components are?

If $\boldsymbol{f}:X\to\mathbb{R}^n$ is a $\mu$-measurable function, I think it is quite easy to see that its components $f_i$ also are. In fact, the projection $\pi_i:\mathbb{R}^n\to\mathbb{R}$, $\...
4
votes
2answers
89 views

Is boundedness required in equivalence between $\frac1n\sum_{k=1}^na_k\to0$ and $\frac1n\sum_{k=1}^na_k^2\to0$?

Suppose $a_n$ is a sequence of non-negative real numbers. If $a_n$ are un-bounded, then I want to know if $\dfrac{1}{n}\sum_{k=1}^na_k\to0$ as $n\to\infty$ is equivalent to $\dfrac{1}{n}\sum_{k=1}^...
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0answers
24 views

Prove that $0\leq \varphi_n\leq\varphi_{n+1}\leq f$ and that $0\leq f-\varphi_n\leq 2^{-n}M$.

The following is from Carothers' Real Analysis: Suppose $f$ is a nonnegative, bounded, Lebesgue measurable function on $[a,b]$ with $0\leq f\leq M$. Let $E_{n,k}=\left\{\frac{kM}{2^n}\leq f < \...
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2answers
62 views

Sum involving $\cosh$ and $\sinh$

I would like to prove the equation $$\frac{\sinh\left(\left (1-\frac{1}{2m} \right)x\right)}{\sinh(x/2m)}=1+ \sum\limits_{n=1}^{m-1}2\cdot \cosh\left(\left( 1-n/m \right)x\right),\quad \forall x > ...
2
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0answers
33 views

Convergence of sequence with $\zeta$ function

Last time I heard interesting question. Unfortunately I do not have idea how to solve it, so I decided to give it here. Let us define sequence $a_n=(\underbrace{\zeta\circ...\circ \zeta}_{n})(\pi)$ ...
0
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1answer
33 views

Limit of the integral of $f(\epsilon t) g(t)$ as $\epsilon\to 0$, when $f$ has bounded variation

I am needing such a "result", which I do not know if it is true, for another question, Convolution with Gaussian question. Let $f\in L^\infty(\mathbb R)$ be of bounded variation on any interval $[a,b]...
2
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0answers
22 views

Splitting the region and estimating fractional Sobolev norms

I've been reading the paper "On the Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces" by Maz'ya and Shaposhnikova and struggling with the short style ...
1
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1answer
50 views

how can I show that $F(x)=\int_0^x f(t)dt$ is continuous on $[0,1]$?

Let $f\in L^1(0,1)$. For $x\in [0,1]$, we define $$F(x)=\int_0^x f(t)dt.$$ How can I show that $F$ is continuous ? What I tried is $$\lim_{h\to 0}F(x+h)-F(x)=\lim_{h\to 0}\int_x^{x+h}f(t)dt=\int_x^x ...
0
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2answers
44 views

Prove that $\frac{1}{2} (e^R - e^{-R}) \geqslant \frac{1}{4} e^R $ with $R > 0$

Sorry to bother you with silly question, but I can't figure out how to prove: $$\frac{1}{2} (e^R - e^{-R}) \geqslant \frac{1}{4} e^R $$ with $R > 0$. I tried different ways but that didn't lead ...
-1
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1answer
30 views

Differentiablity of distance function.

Take the closed interval $[0,1]$ and open interval $(1/3,2/3).$Let $K=[0,1] \(1/3,2/3).$ For $x\in[0,1]$ define $f(x)=d(x,K)$ where $d(x,K)=$inf$\{$ |x-y|:y $\in$ K}. Then $1. f:[0,1]\rightarrow\...
2
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3answers
77 views

Convergence of $\sum a^{1/x_n}$ for $a$ in $(0,1)$ and $\sum x_n$ a positive convergent series

Let $\sum x_n$ be a convergent series of positive real numbers and $0<a<1 $, then is the series $\sum a^{1/{x_n}}$ convergent ? I have only figured out that $\lim a^{1/{x_n}}=0$.
1
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1answer
42 views

Prob. 2, Sec. 2.8 in Kreyszig's functional analysis text: What is the norm of these bounded linear functionals on $C[a,b]$?

Let $C[a,b]$ denote the normed space of all the continuous (real or complex-valued) functions defined (and continuous!) on the closed interval $[a,b]$ on the real line, where $a, b \in \mathbb{R}$ and ...
2
votes
1answer
66 views

Is a limit a formalized infinitesimal?

From what I understand after thinking about this, delta epsilon really seems to formalize the notion of an infinitesimal. The constraint $0<|\delta-c|$ combined with the fact that there is no real ...
0
votes
1answer
49 views

Find the largest possible radius for $S=[1,4) \cup (4,9]$

Find max $\{\epsilon : N(x;\epsilon) \subseteq S\}$, the largest $\epsilon$ such that the neighborhood centered at $x$ of radius $\epsilon$ is contained in $S$. That is, state the largest possible ...
0
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1answer
28 views

A specific maximal function of of a potential function

Let $$f(x)=\frac 1{(1+|x|)^2},$$ Then what's the maximal function of $f$ ? By definition $$Mf(x)=\sup_{r>0}\frac 1{|B_r(x)|}\int_{B_r(x)}\frac 1{(1+|y|)^2}dy,$$ If one can prove that the average ...
1
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0answers
31 views

A vector field in a star shaped set

I'm having problems trying to proof Poincaré's lemma for Star-Shaped sets Let $F:U\to \mathbb R^{2}$ be $C^{1}$ functions,where $U\subset \mathbb R^{2}$ is a Star-shaped set. If $F_x:U\to \mathbb ...
3
votes
2answers
67 views

Is there a contradiction between these two concepts in Cauchy sequence?

I'm studying real analysis from Rudin's Principles of mathematical analysis, and I feel confused about some definition in Cauchy sequences and feel like there is a contradiction in some idea, or maybe ...
1
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4answers
76 views

prove triangular inequality for $ d(x,y)= \frac{||x-y||}{1+||x-y||}$ [duplicate]

prove triangular inequality for $$ d(x,y)= \frac{||x-y||}{1+||x-y||}$$ that is $d(x,y) \leq d(x,z)+d(z,y)$ ofcourse ||.|| is a norm and has properties of norms this usually works $$ \begin{...
3
votes
4answers
36 views

How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$?

Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$. One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$...
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0answers
24 views

Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

This question has been post in here. Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable). ...
3
votes
1answer
71 views

Functional inequality $\sum_{1\le i<j\le n}f(x_i+x_j)\ge \frac{n(n-1)}{2}f(a_1x_1+a_2x_2+…+a_nx_n)$

Let $n\in\mathbb N, n\ge 2$. Does there exist a set of non-zero real numbers $a_1, a_2,..,a_n$ with this condition: If function $f: \mathbb R \rightarrow \mathbb R$ satisfies the inequality $$...
1
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3answers
42 views

Differentiability of piecewise functions

Check whether the function is differentiable: $$f:\mathbb{R}^2\rightarrow \mathbb{R}$$ $$f= \begin{cases} \frac{x^3-y^3}{x^2+y^2} & (x,y)\neq (0,0) \\ 0 & (x,y) = (0,0) \\ \end{...
0
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1answer
26 views

Power series examples with different properties

I'm trying to accumulate some examples of power series with certain properties according to an exercise in Abbott's analysis text. I'm having trouble developing examples for the following: 1) power ...
0
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0answers
34 views

Small-Oh proofs

I'd like to prove $\ o(x^{m+n}) \subseteq o(x^m)*o(x^n) $. I'd like to prove by definition. Let $\ f(x) \in o(x^{n+m})$ and $\ h*g=f $ $\ \lim_{x \to \infty}\frac{f(x)}{x^{m+n}}=0 $ From this we ...
1
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1answer
80 views

existence and uniqueness solution Cauchy problem $y'=y(y+1)e^{-y}$

I have this Cauchy problem ($\alpha \in \mathbb{R}$) \begin{cases} y'(t) = y(y+1)e^{-y} \\ y(0) = \alpha \end{cases} $f(t,y)=y(y+1)e^{-y} \in C^1(\mathbb{R}^2)$ so for Cauchy-Lipschitz theorem ...
1
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1answer
49 views

Inner product on $\mathbb{R}[X]$

Let $P$ and $Q$ be two polynomials in $\mathbb{R}[X]$ and let $$\langle P,Q\rangle =\int _{-\infty}^{+\infty}P(x)Q(x)f(x)dx$$ with $f(x) = \frac{1}{\sqrt{2\pi}} \exp(-x^2/2)$. I would like to ...
0
votes
1answer
34 views

Condition expectation of functions: $E(fg\mid\mathcal{A})=gE(f\mid\mathcal{A})$ when $|g|<\infty$ a.e.

Let $(X,\mathcal{B},\mu)$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ a sub-$\sigma$-algebra, then by an easy application of the Radon-Nikodym Theorem, letting $\nu(A)=\int_A\, f\,\mathrm{...
0
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1answer
34 views

Good reference for partitions of unity?

I am reading about Sobolev Spaces and regularity theory of PDEs. The partition of unity lemma, as stated in Haim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations, is as ...
-1
votes
0answers
37 views

Show that any $f\in C^1(\mathbb{R}^n, \mathbb{R}^m)$ is Lipschitz

Show that any $f\in C^1(\mathbb{R}^n, \mathbb{R}^m)$ is Lipschitz. My idea is to use the corollary that follows from the mean value theorem that says that if $f$ is differentiable on some interval $I\...
0
votes
2answers
47 views

Orthogonal of an Hilbert subspace and density

If $V$ is a subspace of an Hilbert space $H$, I know that the orthogonal of $V$, $V$$^o$, is ($V$closed)$^o$, even if $V$ is not closed. Does this mean that $V$ is always dense in $V$$^o$? Thanks!...
2
votes
1answer
64 views

Minimum of a function in $(0,1) \times (0,+\infty)$

I would like to minimize the function $$ (\alpha,\theta) \mapsto F(\alpha,\theta) := -\theta x^\alpha + \sum_{k=1}^N \ln(1+p_k(e^{\theta \ell_k^\alpha}-1)) $$ where $\theta \in (0,+\infty)$, $\alpha \...