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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with 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.

0
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0answers
3 views

Open homogeneous functions are onto.

Let $f:\mathbb{R}^{n}\to\mathbb{R}^{m}, n,m\in \mathbb{N}$ be a continuous function such that $f(\lambda x)=\lambda^{p}f(x)$(homogeneous), $p\geq0$. Show that f is an open map, then f is surjective, ...
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0answers
7 views

Prove that $G$ is differentable on $U$ and $G'(x)=\int^{b}_{a}D_2f(s,x)ds,\;\;x\in U$

Let $U$ be open in $\Bbb{R}^n$ and \begin{align}f:[a,b]\times U\to \Bbb{R}^m\end{align} be continuous. Let \begin{align}G(x)=\int^{b}_{a}f(s,x)ds,\;\;x\in U.\end{align} We assume that $D_2 f$ is ...
0
votes
0answers
10 views

Fixed point of a closed ellipse

Consider the closed ellipse $$E=\{(x,y) \in \Bbb{R}^2: 2x^2+3y^2 \leq 1\}$$ Prove or disprove: Every continuous map from $E$ to $E$ has a fixed point Is Brouwer fixed point theorem useful here? ...
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0answers
9 views

Prove that $\ell(\Bbb{R}^n,\ell(\Bbb{R}^m,\Bbb{R}^q))=\ell_b(\Bbb{R}^n\times\Bbb{R}^m,\Bbb{R}^q)$

Prove that \begin{align}\ell(\Bbb{R}^n,\ell(\Bbb{R}^m,\Bbb{R}^q))=\ell_b(\Bbb{R}^n\times\Bbb{R}^m,\Bbb{R}^q)\end{align} where $\ell(\Bbb{R}^n,\ell(\Bbb{R}^m,\Bbb{R}^q))$ represents the space of ...
0
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0answers
8 views

Let $\Phi:ISO(\Bbb{R}^n)\to ISO(\Bbb{R}^n)$, then prove that $\Phi$ is a $C^1-$function and a $C^1-$diffeomorphism

Let $$\Phi:ISO(\Bbb{R}^n)\to ISO(\Bbb{R}^n)$$ I want to prove that $\Phi$ is a $C^1-$function and a $C^1-$diffeomorphism. I know that $\Phi$ is $C^1-$diffeomorphism if $\Phi$ is $C^1$ $\Phi$ is ...
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0answers
8 views

Problem of metric space comnined with differentiability.

Let $U$ be an open connected subset of $\mathbb{R}^n$. Let $f:U \to \mathbb{R}$ be a differentiable map such that $Df(p)=0$ for all $p\in U$. Prove that $f$ is constant.
2
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0answers
19 views

Given a smooth function $f:\mathbb R^n\rightarrow\mathbb R$ with $f(0)=0$, $f(tx)/t$ is the restriction of a smooth function.

Suppose there is a smooth($C^\infty$) function $f:\mathbb R^n\rightarrow \mathbb R$ with $f(0)=0$. Define a function $F:\mathbb R^{n+1}\rightarrow\mathbb R$ by setting $(t,x_1,\ldots,x_n)\mapsto t^{-1}...
1
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0answers
19 views

The $L^{2}$-difference of a Lipschitz function and its mollified function.

Question Let $f$ have Lipschitz constant at most $2$, and $\varphi$ be a standard mollifier. That is: $\varphi$ is smooth and has support contained in $B(0,1)$ and satisfies, $$ \begin{cases} 0 \le \...
1
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1answer
43 views

Quotient Group $U(1)/\mathbb{Z} $

Define a homomorphism $f_t$ from $\mathbb{Z}$ to $U(1)$ by $$f_t: n \rightarrow \exp {(i2\pi n t)},\,\,\,\,\,\, n\in \mathbb{Z}$$ where $t\in[0,1)$. Obviously if $t=1/m$ with $m\in\mathbb{Z}^+$, the ...
1
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0answers
14 views

Write explicitly a parameterization for $\Sigma \cap V$ with $M \in V$ (application of the Implicit Function Theorem)

(a) State the Implicit Function Theorem in the most general way that you know (b) Let $\Sigma$ the set of $2 \times 2$ matrices with determinant zero. Show that if $0 \neq M \in \Sigma$, then ...
7
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3answers
57 views

Pointwise convergence of sequence $(f_n)_n$ of functions to $f$ and changing limits

My analysis notes contains the following question: if $(f_n)_n$ is a sequence of functions of $A \subset \mathbb{R} \to \mathbb{R}$ and $a \in \mathbb{R} \cup \{-\infty, +\infty\}$ an accumulation ...
1
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2answers
35 views

Prove formally that multivariate function has global extrema

Consider the following function: $f(x_1, \dots, x_n) = p_1 \log x_1 + \dots + p_n\log x_n$ subject to the constraint that $\sum_i p_i = \sum_i x_i = 1$. It is also known that $p_i \in [0,1]$ and $x_i \...
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0answers
37 views

If $f(b)=\max\left|\sin x+\frac{2}{\sin x+3} +b\right|$, what is $\min f(b)$? [on hold]

For any real number $b$ let $f(b)$ denote the maximum of: $$ \left|\sin x+\frac{2}{\sin x+3} +b\right|$$ for all $x$ belongs to $\mathbb R$. Then the minimum value of $f(b)$ for all $b$ belongs to $\...
1
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2answers
38 views

Showing the set with a supremum has an increasing sequence converging to that supremum.

let $A$ be an infinite subset of $\mathbb R$ that is bounded above and let $u=\sup A$. Show that there exists an increasing sequence $ (x_n) $ with $x_n \in A $ for all $n\in \mathbb N$ such that $u ...
0
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0answers
21 views

Tight upper bound for an improper integral

I need to find a more tight upper bound for the improper integral $\int_0^{\infty}e^{-ax-x^b}dx$ rather than the following approximation based on Jensen's inequality $\mathbb{E}(f(X))<f(\mathbb{E}(...
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0answers
15 views

How to evaluate analytically the inverse Laplace transform of the following parameterized expression?

I am trying to evaluate analytically the inverse Laplace transform of the following expression: \begin{equation} F(p) = \frac{1}{ p^2 \left[ 1 + \alpha S_p^{-1/2} \tan \left( \frac{\pi}{2} S_p^{-1/2} ...
2
votes
1answer
51 views

Proof verification that every compact set has a finite subcover

Let $S$ be a (possibly uncountable) set of open sets, and $E$ be a compact set with $E \subset \mathbb{R}$ such that $\displaystyle E \subset \bigcup_{O \in S} O$. Then there's a subset $T \subset S$ ...
3
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0answers
40 views

How do not generate a $\sigma$-algebra

Let $\mathscr{C}\subset \mathscr{P}(\Omega)$ be a class of subsets of a nonempty set $\Omega$ containing $\Omega$ and $\varnothing$. Define $\mathscr{C}_0=\mathscr{C}$ and for each $n\geq 1$ define ...
1
vote
2answers
33 views

Convergence of $\sum_{n=1}^{\infty}\left(\frac{n}{n^2+1}\right)^{k(n)}\,\,\,\,\,;\,\,k(n)=\frac{1}{\cos\left(\frac{1}{\ln^{a}(n)}\right)}$

Study the convergence of the series as $a > 0$ $$\sum_{n=1}^{\infty}\left(\frac{n}{n^2+1}\right)^{k(n)}\,\,\,\,\,;\,\,k(n)=\frac{1}{\cos\left(\frac{1}{\ln^{a}(n)}\right)}$$ In the text there's ...
1
vote
1answer
33 views

Question about a nowhere dense set define in terms of spheres

Suppose that the set $A \subset \mathbb{R}^2$ can be written as follows: \begin{align} A= \bigcup_{i \in I} S_i \end{align} where \begin{align} S_i =\{ x \in \mathbb{R}^2: \|x\|=c_i \} \end{...
3
votes
1answer
80 views

If $f$ is zero except at finitely many points, then $\int_{a}^{b} f = 0$

Let $f: [a,b] \to \mathbb{R}$ be zero everywhere except at the points $s_1, \dots s_k$. Prove that $f$ is integrable and $\int_{a}^b f = 0$, directly from the definition. Attempt: Let $\epsilon > ...
1
vote
1answer
67 views

Proof explanation: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$ [duplicate]

Two days ago, I asked a question Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$ but was answered just once. However, I am finding it ...
1
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1answer
46 views

Using the Mean Value Theorem, prove this inequality.

Using the Mean Value Theorem, prove that $|\cos^2(b)-\cos^2(a)|\gt \frac{1}{4}|b-a| $ for all $a,b \in (\frac{\pi}{4},\frac{\pi}{3})$ So far, I have Let $f(x)=\cos^2(x)$ Then $f(x)$ is ...
4
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2answers
446 views

Real Analysis sequence limit problem

Sorry for the unclear title, the problem is too specific so I couldn't think of anything else. Here goes: If Prove that: Now, in my textbook there is a proof provided but I don't understand it. ...
0
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0answers
15 views

Prove that probability function $Pr(v)=\int_{G(v)} dP$ is smooth

There is a probability density function that depends on non-deterministic ($v$) and random ($x$) parameters: $Pr(v)=\int_{G(v)} dP$, where $G (v)$ is the "goal" region, the probability of getting ...
0
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1answer
13 views

I want to find A real valued function having a continuous first derrivative for all points in domain, but with undefined higher order derivatives.

A real valued function having a continuous first derrivative for all points in domain, but higher order derivatives are not even defined Is there an example of such a function?
0
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1answer
21 views

Proving differentiability of a function

We define the function $f$ in the following manner: $$f(x) = \begin{cases} 0, & \text{if $x = 0$} \\ \vert x \vert^\alpha \text{sin}(\frac{1}{x}), & \text{if $x \neq 0$} \end{cases}$$ Prove ...
0
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0answers
34 views

Prove that $f : \mathbb{R} \rightarrow \mathbb{R}$ is lower semi-continuous

Prove that $f : \mathbb{R} \rightarrow \mathbb{R}$ is lower semi-continuous if and only if if the set$ \{(x,y) \in \mathbb{R}^2 : y \ge f(x) \}$ is closed in $\mathbb{R^2}.$ My Proof : $f$ ...
3
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2answers
37 views

Example for Sufficiency of Conditions for Differentiating Under Lebesgue Integral

In the book I am reading*, sufficient conditions for differentiating under the Lebesgue integral are stated as follows: $J\in \mathbb{R}$ is an interval and $\left(X,\mathcal{M},\mu\right)$ a ...
4
votes
1answer
21 views

Checking the range in which the given function is uniformly continuous

Problem For $p \in \mathbb{R}^1 $, let $f(p) = (2p+1,p^2)$. a) Prove that $f:\mathbb{R}^1 \to \mathbb{R}^2 $ is uniformly continuous on the closed interval $[0,2]$. b) What is the largest interval ...
3
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2answers
41 views

Showing the convergence of the given recursively defined sequence

Let $(a_1) = 1$ and $$(a_{n+1})= \frac{4+3a_n}{3+2a_n}$$ It is required to show that the following recursively defined sequence converges. I know one way to show this converges. Define $$f(x)= \frac{4+...
1
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0answers
44 views

Convergence or divergence of the improper integral [on hold]

We have an arbitrary function $u(x)$. It is known that $u(x)$ continuous and monotonically increasing in $[0,\rho]$, $u(0)>0$, $1-\delta\,u(\rho)=0\,(\delta>0)$, $1-\delta\,u(0)>0$ and $u'(\...
1
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0answers
38 views

Interchangeable limits

The following exercise 2.2.9 is borrowed from Terence Tao's Analysis II, page 33. While I understand the problem completely, I lack the technique to attack it. So I will really appreciate a small hint ...
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0answers
24 views

How to derive Riesz transform from its Fourier transform?

Here I encounter a problem when I want to derive Riesz transform: For $f(x)\in\mathcal{L}^p,1<p<\infty.$ We know the Fourier transform of Riesz transform: $$\widehat{R_jf}=\frac{\xi_j}{|\xi|}\...
0
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1answer
54 views

$f: \mathbb R^2 \to \mathbb R$ be $f(x,y) = x^{[y]}$. How can we define this function at $(0,0)$?

For any $y \in \mathbb R$, let $[y]$ denotes the greatest integer less than or equal to $y$. Define $f: \mathbb R^2 \to \mathbb R$ be $f(x,y) = x^{[y]}$. Then there are four options from which any ...
1
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1answer
25 views

How to check a function whether it is of bounded variation or not in an open interval.

How to check a function whether it is of bounded variation or not in an open interval. $f(x) = \sqrt{(1 - x^2)}$ Is this function of bounded variation on $x \in (-1,1)$? My thought: I know how ...
3
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0answers
62 views

Convergence of $\sum_k^{n-1}f\left(\frac{k+\theta} n\right)-n\int_0^1 f(x)\,dx$ doubt

Let $\theta\in[0,1]$ be a constant and $f\in C^1[0,1] $. Show that $$\sum_k^{n-1}f\left(\frac{k+\theta} n\right)-n\int_0^1 f(x)\,dx$$ converges to $(\theta-\frac{1}{2})(f(1)-f(0))$, as $n\to\...
4
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1answer
42 views

Convergence of$\int_{0}^{+\infty}\frac{1}{\left(\log(x)\right)^{4\alpha}}\sin^2\left(\frac{1}{x^{\alpha}}\right)\,dx$

Study the convergence of the following integral as $\alpha >0$ $$\int_{0}^{+\infty}\frac{1}{\left(\log(x)\right)^{4\alpha}}\sin^2\left(\frac{1}{x^{\alpha}}\right)\,dx$$ We have to study the ...
1
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4answers
44 views

Show convergence and find the limit of the sequence given by $a_1=1$ and $a_{n+1}=\frac{1}{3+a_n}$

I've been trying to solve this exam question on an exam in real analysis. Thus, only such methods may be used. The problem is as follows. Show that the sequence $a_n$ defined by $a_1=1$ and $a_{n+1}...
2
votes
1answer
31 views

Showing that a polynomial is zero given that a sum containing its coefficients sum to zero

I've been trying to solve this exam question on an exam in real analysis. Thus, only such methods may be used. The problem is as follows. Assume that $c_0,c_1,\dots,c_n$ are real numbers so that $$\...
1
vote
1answer
53 views

Does the series $\sum_{k=1}^{\infty} (\sin \frac{1}{k} - \arctan\frac{1}{k})$ converge?

I am shockingly terrible at determining whether or not infinite series converge or not... I'm stuck on the problem: Does the series $\sum_{k=1}^{\infty} (\sin \frac{1}{k} - \arctan\frac{1}{k})$ ...
0
votes
3answers
84 views

If $x'(t)$ is bounded for $x\geq0$ show that $\lim_{t\to\infty}{x(t)}=0$

Assume that $x(t)$ is nonnegative for $t\geq0$ and $\int_0^\infty{x(t)\;dt}<\infty$. If $x'(t)$ is bounded for $x\geq0$, then show that $\lim_{t\to\infty}{x(t)}=0$. I started with contradiction ...
0
votes
0answers
45 views

No Accumulation Point sin(nx)

I hope this question is not too trivial - when I was solving a different problem, I somehow strived off and eventually conjectured this. Let $(V, \| \cdot \|)$ be a normed space and let $r > 0$. ...
2
votes
2answers
44 views

How Does a Fourier $\sin$/$\cos$ Series Arise From a “Normal” Fourier Series? How Does This Relate to the Generalised Fourier Series?

I am told that Fourier showed that we can represent an arbitrary continuous function, $f(x)$, as a convergent series in the elementary trigonometric functions $$f(x) = \sum_{k = 0}^\infty a_k \cos(...
1
vote
3answers
45 views

Where do we use continuity?

If $f$ is continuous on $\mathbb{R}$, $f'(0)=1$ and $f(x+y)=f(x)f(y)$ for all $x \in\mathbb{R}$, show that $f'(x)=f(x)$ for all $x\in\mathbb{R}$. Solution: It is clear that $f(0)=1$. For each $x$ we ...
0
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0answers
22 views

Application of Higher Order Derivatives

If $V = \mathbb{R}^{n}$ and $W = \mathbb{R}^{m}$, and $f:V\rightarrow W$ is smooth, then higher order derivatives at $x\in V$, $D^{n}_{x}f$ can be thought of as symmetric, multilinear functions $V\...
1
vote
1answer
21 views

Uniformly convergence of a sequence defined in compact set.

Let $(f_{n})$ be a sequence of functions $[-1,1] \to \mathbb{R}$ defined by: $f_{0}(t)=0$ and $f_{n+1}(t)=f_{n}(t)+\frac{1}{2}(t^{2}-f_{n}^{2}(t))$. Show that $(f_{n})$ converges uniformly. ...
1
vote
3answers
59 views

How to prove the Bijection of an Interval $(-1,1)$ to $\mathbb{R}$?

How can I prove that this image is bijective? $$f: (-1,1) \longrightarrow \mathbb{R}, \quad x \longmapsto \frac{x}{1-x^2} $$ is bijective without the use of the Steepness of the slope?
1
vote
2answers
63 views

Boundedness of a sublevel set of a convex function

Let $f : \mathbf R^n \to \mathbf R$ be a convex function with $f(0)=0$ and $\displaystyle\lim_{t\to\infty}f(tx)=\infty$ for any $x\in\mathbf R-\{0\}$. Is $$A:=\{x \mid f(x)\le 1 \}$$ bounded? I was ...
3
votes
1answer
56 views

About $f:[0,1]\rightarrow \mathbb{R}$ integrable with $f \geq 0$, and $A_{t}=\{x\in [0,1]\mid t \leq f(x) \leq 2t\}$

The question goes like this: Let $f:[0,1]\rightarrow \mathbb{R}$ be an integrable (lebesgue), non-negative function. We denote for all $t>0$, the set: $A_{t}=\{x\in [0,1]\mid t\leq f(x) \leq 2t\}$...