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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.

4
votes
1answer
62 views

Can solution to differential equation be continuous extended?

Let $v:\mathbb{R}^n\to\mathbb{R}^n$ be continuous. Let $\gamma:[0,T)\to\mathbb{R}^n$ be the unique solution to $\frac{d\gamma}{dt}=v(\gamma),\gamma(0)=p$. Suppose $\gamma$ remains bounded. Show that $\...
10
votes
1answer
257 views

“Counterexample” for the Inverse function theorem

In my class we stated the theorem as follows: Let $\Omega\subseteq\mathbb{R}^n$ be an open set and $f:\Omega\to\mathbb{R}^n$ a $\mathscr{C}^1(\Omega)$ function. If $|J_f(a)|\ne0$ for some $a\in\Omega$...
0
votes
2answers
40 views

Applying Cantor's Diagonal Argument to natural number, it gives us a new number different from all natural numbers. What is it?Is it infinity? [on hold]

Here's the way to generate a number using Cantor's Digonal Argument: First, present natural number set as the following form: $a_1a_2a_3......a_n$ $b_1b_2b_3......b_n$ $c_1c_2c_3......c_n$ This ...
0
votes
1answer
31 views

What does it mean to be analytic on a compact set

For example, what does it mean for $f:[-1,1] \to \mathbb{R}$ to be analytic on $[-1,1]$? For $x \in (-1,1)$ I assume that it means there exists an interval $(x-\delta, x+\delta)$ such that the Taylor ...
0
votes
1answer
33 views

Trouble understanding a point of accumulation as a limit superior and understanding its properties

Basically, I am having trouble understanding this hole page from my complex analysis book. I fail to understand why a point of accumulation of a given set is also its limit supperior and most ...
5
votes
0answers
75 views
+50

Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$

How Can I evaluate the following sum$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos(\frac{m\pi x}{a})=\frac{\cos(\lambda_0 x)}{\sin(\lambda_0 a)}$$...
0
votes
1answer
47 views

(Dis)prove continuity, integrability of $f:[0,1] \rightarrow \mathbb{R}$ with 'jumps'.

Let $f:[0,1] \rightarrow \mathbb{R}$ be $f(x)= \begin{cases} x \text{ if } x \neq \frac{1}{n} \text{ for all } n \in \mathbb{N} \\ 1 \text{ if } x =\frac{1}{n} \text{ for some } n \in \mathbb{N}...
2
votes
1answer
44 views

Prove that $\sin(ax+b)$ is continuous using $|\sin x| < |x|$

Prove that $\sin(ax+b)$ is continuous using $|\sin x| < |x|$ I would like to ask for verification of my proof. Essentially to obtain a proof we need to show that: $$ \lim_{x\to x_0}\sin(ax+b) = \...
0
votes
0answers
47 views

Existance of concave balanced function

We know that the core of a balanced game is non-empty. The convexity ensures balancedness. However, I was wondering if a concave function too satisfies balancedness condition. The balancedness is ...
3
votes
2answers
65 views

Dirichlet problem with odd function.

Let $\Omega \subset \mathbb{R}^2$ be open, bounded and symmetric with respect to the origin. Let$ f:\partial \Omega \to \mathbb{R} $ be odd and continuous function. Show that if $u$ is solution of: $$...
3
votes
1answer
25 views

$\mathcal{C}^r$ topology in the germ space.

I'm reading the article "Local and simultaneous structural stability of certain diffeomorphisms - Marco Antônio Teixeira", and on the first page says "Denote $G^r$ the space of germs of involution at $...
0
votes
2answers
45 views

Prove that there exists a function $ g: [0,1] \to \mathbb{R} $ which is 1-Lipschitz, satisfies $ g (0) = 0 $ and has the following property.

Question. Consider a continuous function $ f: [0,1] \to \mathbb {R} $. Prove that there exists a function $ g: [0,1] \to \mathbb{R} $ which is 1-Lipschitz, satisfies $ g (0) = 0 $ and has the property ...
2
votes
1answer
23 views

Dominating function for derivative of moment generating function

Let $X$ be a random variable and the moment generating function $$\psi_X:(-\varepsilon,\varepsilon)\rightarrow \mathbb{R}_+,\quad \psi_X(t):=E[e^{tX}]$$ be defined, such that $\psi_X(t)<\infty$ ...
0
votes
0answers
28 views

Motion of a particle defined via Lebesgue integral

Suppose $x_0 =0$. A particle moves as follows: $$x_t = \int_0^t a(s) ds$$ where $a: \mathbb{R}_+ \to \{-1,0,1\}$ is a measurable function. Suppose, I have that $a_s = 1$ if $x_s = 0$. I want to ...
1
vote
3answers
43 views

Why are polynomials in $[0,1]$ not isomorphic to $C[0,1]$.

I'm learning about Stone-Weisstrass Theorem and I learn that the closure of the polynomials is $C[0,1]$, I don't know where I read that they the polynomials in $[0,1]$ where isomorphic to $C[0,1]$. ...
-1
votes
1answer
118 views

Calculate $\lim_{n\to\infty}\frac{1}{2^n}\sum_{k = 1}^{n}\tfrac{1}{\sqrt{k}}\binom{n}{k} $

$$\lim_{n\to\infty}\frac{1}{2^n}\sum_{k = 1}^{n}\left(\frac{\binom{n}{k} }{\sqrt{k}}\right)$$ Since $\frac1{2^n}\binom{n}{k}$ approximates a normal distribution with mean $\frac n2$ and variance $\...
0
votes
0answers
27 views

Showing that if $f = 0$ whenever $f=0$ on $\Omega \subset X$ then $\Omega$ is dense in $X$

I'm trying to solve this exercise in the lecture notes for a course in analysis that I'm taking. Proving that that defines a seminorm is straightforward. I'm only stuck in the step where I have to ...
0
votes
0answers
23 views

Comparing two curves

For my Bachelor thesis, I have to analyse the goodness of an Approximation. The exact formula is called $\mathcal{I}(\varepsilon)$ and the Approximation $\mathcal{I}_{approx}(\varepsilon)$. In order ...
1
vote
1answer
42 views

Confusion about a calculation concerning weak and strong convergence

I'm reading something and the consider the following example: Define: $$\chi(x) = \mathbb{1}_{[0,1/2]}-\mathbb{1}_{[1/2,1]}$$ and extend periodically to $\mathbb{R}$. Let $u_j = \chi(jx)$ on $(0,1)$...
0
votes
1answer
22 views

subset of a special type of a set [on hold]

Any help is appreciated. Can't process the hint.
0
votes
3answers
50 views

Limit of a sequence as $n\to\infty$ [on hold]

Let $x_{0}$ be a positive real number and $n\in\mathbb{N}$. Then what is $$ \lim_{n\to\infty}\{(x_0+n)^r-n^r\} $$ where $r\in (0,1)$ is fixed number.
0
votes
0answers
29 views

Difficult problems between triangle and square [on hold]

Problems : ABCD is a square centered at O. Let M be a point on the line segment [BD], and P and Q be the perpindicular projections of it on [AB], and [AD] respectively Prove that the lines (CM)...
1
vote
1answer
43 views

Proving the divergence of $\sum_2^\infty{\frac{1}{n \log n}}$ using comparison test

It is quite straightforward using the Cauchy Condensation test. But is there any way to solve this problem using some well known comparison test? I cannot think of any way of my own. Any help/hint ...
0
votes
0answers
72 views

What is $b_1$ if $a_{n+1}+b_{n+1}=\frac{1}{2}(a_{n}+b_{n})$ and $a_{n+1}b_{n+1}=\sqrt{a_{n}b_{n}}$?

Let two sequence $\{a_{n}\},\{b_{n}\},n=1,2,\ldots$ be such that $$a_{n+1}+b_{n+1}=\frac{1}{2}(a_{n}+b_{n})$$ and $$a_{n+1}b_{n+1}=\sqrt{a_{n}b_{n}}$$ If $a_{10}=1$ and $b_{1}>0$, what is $...
1
vote
1answer
36 views

How to understand the convergence of Fourier Series in $L^p$

My professor told me that Suppose that $f \in L^p(-\pi, \pi)$ (i.e. $f$ is 2$\pi$-periodic and $\|f\|_{L^p} < \infty$). If $1<p<\infty$, then the Fourier series of $f$ converges to $f$ in $...
0
votes
0answers
30 views

When can we change the order of improper integral? [on hold]

Let $f(x,y)$ be continuous on $Q=\{(x,y)\mid x>0,y>0\}$ and $\iint_Q |f(x,y)|$ converge. Suppose $$\int_{0}^{\infty} f(x,y)\, \mathrm{d}y$$ converges uniformly on any finite interval. Show that ...
1
vote
2answers
90 views

What is the smallest $n$ such that $\frac{n(n+1)(2n+1)}{6}$ is a square number? [on hold]

Question : Find the smallest natural number $n>1$ such that $\sum_{k=1}^{n}k^2$ is a square number Recall that : $\sum_{k=1}^{n}k^{2}=\frac{n(n+1)(2n+1)}{6}$
0
votes
3answers
27 views

Root multiplicity for non-polynomial function

I know that if $f$ is a polynomial and $f(a)=0,f'(a)=0...,f^{(k)}(a)\neq 0$, then $a$ is a root of multiplicity $k$. Does this work for a differentiable function that is not a polynomial? I have seen ...
1
vote
1answer
49 views

How would I start this problem?

Let $f \in C^2(\mathbb{R})$, let $v\in\mathbb{R}^n$ and $c>0$. Define a function $$ u : \mathbb{R}^n\times \mathbb{R} \to \mathbb{R}, \qquad u(x,t) = f(v\cdot x - c\|v\|t) $$ and show that ...
0
votes
0answers
55 views

On the continuity of an extended continuous function

Let $f$ be a continuous function on a closed subset $F$ of $\mathbb{R}$. We can extend $f$ continuously to a function $g:\mathbb{R}\longrightarrow\mathbb{R}$ in the following way: Let us define $\...
1
vote
0answers
36 views

Functions of Bounded Variation Have Left and Right Limits

I have a proof of this for a real valued function https://www.encyclopediaofmath.org/index.php/Function_of_bounded_variation#Generalizations based on the Jordan decomposition into monotonic functions. ...
2
votes
1answer
28 views

If $X:Ω→\mathbb R^d$ and $Y:Ω→\{0,1\}$, then $\text E\left[\left|\text P\left[Y=1\mid X\right]-Y\right|^2\right]≤\text E\left[|f(X)-Y|^2\right]$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $d\in\mathbb N$, $X$ be a $\mathbb R^d$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$, $Y$ be a $\left\{0,1\right\}$-...
-2
votes
2answers
66 views

Prove that if a sequence is Cauchy then there exists a sub sequence such that $|a_{n_{k+1}} - a_{n_k}| < 1/k^2$ [on hold]

I have to show that if a real sequence $\{a_n\}_{n\in\mathbb{N}}$ is Cauchy then there has to be a sub sequence $\{a_{n_k}\}_{k\in\mathbb{N}}$ satisfying the following: $$\left|a_{n_{k+1}}-a_{n_{k}}\...
2
votes
4answers
54 views

for which $a \in \mathbb{R}$ does the series $\sum_{k=1}^\infty \frac{1}{k^a}$ converge

I want to show for which $a \in \mathbb{R}$ the series $\sum_{k=1}^\infty \frac{1}{k^a}$ converges. For $a = 0$ the series diverges for $a < 0$ we have $\frac{1}{k^{-a}} = k^a$ and the series ...
0
votes
2answers
32 views

Example of $\sum a_n$ convergent but $\sum \sqrt{a_{2^k}}$ divergent?

The problem is this: Prove that the convergence of $\sum a_n$ implies the convergence of $$\sum \frac{\sqrt{a_n}}{n},$$ if $a_n\geq 0.$ I'm using the Cauchy's method: $\sum \frac{\sqrt{a_n}}{n}$ ...
0
votes
1answer
22 views

Uniform convergence- Point-wise convergence. Doubt regarding the difference

I know the definition of "pointwise convergence" and "uniform convergence", nevertheless I have some difficulties understanding the difference between those two concepts. My book defines Uniform ...
1
vote
1answer
24 views

Why does calculating the quadratic variation of a Brownian motion in this way not work?

This seems like a simple question, but I am stumped. I know the proofs for quadratic variation and cross variation, etc..., but for some reason can't understand why the following doesn't make sense to ...
2
votes
2answers
70 views

$ \int_{-\infty}^\infty f(x)e^{-2\pi i\xi x}dx=0$ for all $\xi\in\mathbb R$ with $|\xi|>M $

Suppose that $f(z)$ is an entire function on $\mathbb C $, and that there exist constants $M>0$ and $A>0$ such that $$ |f(x+iy)|\le\frac{A}{1+x^2}e^{2\pi M|y|}\quad \text{for all}\ x,y\in\...
0
votes
0answers
17 views

Given a probability density function $p$, find a factorized distribution $q$, such that $\frac{p(x)}{q(x)}$ is bounded above?

Given a bounded, continuous probability density function $p$ over $\mathbb{R}^n$, is it always possible to find a factorized distribution $q$, $q(x) = \prod_{i=1}^n q(x_i)$, so that $\frac{p(x)}{q(x)}$...
4
votes
3answers
63 views

Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function

This question was asked in ISI BStat / BMath 2018 entrance exam: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that for all $x\in\mathbb{R}$ and $t\ge 0$, $$f(x)=f(e^{t}x)$$ ...
3
votes
2answers
130 views

Integral $\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$

Prove that $$\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$$ I was given this integral in my post Request for crazy integrals. I have never seen an integral like this before ...
0
votes
2answers
22 views

Does $\exists f'(x_0)$ imply that $f$ is defined on $N(x_0, \epsilon)$ for some $\epsilon>0$?

We define the derivative at $x_0$ as, For $f : (a,b) \to \mathbb{R}$, and $x_0 \in (a,b)$ $$f'(x_0) \overset{def}{=} \lim_{h\to 0} \frac{f(x_0 + h) - f(x_0)}{h}$$ Question So, we often use ...
0
votes
3answers
58 views

Is $\mathcal O(n,\mathbb C)$ Compact?

I know that $\mathcal O (n,\mathbb R)$ is compact. One can prove easily by using the continuous map $det(A),$ and norm is bounded. By Heine -Borel theorem it is compact. But I got stuck to prove it ...
0
votes
1answer
56 views

If $f'=[f]^2$ and $f(0)=0$, what we can say about $f$?

Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable and $f(0)=0$ s.t. $\forall x \in \mathbb{R}$ we have $$f'(x)=[f(x)]^2,$$ where $[x]$ is the least integer greater than or equal to $x.$ Show ...
3
votes
2answers
44 views

Assumption on Mirror Descent convergence?

I am studying Mirror descent and nonlinear projected subgradient methods. At page 171 Theorem 4.1., the author claims that the method converges provided $$ \sum_s t_s= \infty , \,\,\,t_k \rightarrow ...
0
votes
1answer
15 views

nested, closed, bounded sets in complete metric space is non empty

I am self-studying Real Analysis and doing the exercise 3.21 of Rudin, Principles of Mathematical Analysis. I got the same proof idea, but cannot reach the argument that ``$x$ must belong to $E_n$ ...
1
vote
2answers
71 views

Help showing $\sin(x)^{\sin(x)} < \cos(x)^{\cos(x)}$ between $x = 0$ and $x = \frac{\pi}{4}$?

This is a homework problem that I just can't make progress on. I know that it can be shown through a special case of the generalized arithmetic-geometric mean inequality ($a^t b^{1-t} \leq ta + (1-t) ...
2
votes
2answers
43 views

Partial fraction expansion of $\frac{1}{(s+1)^{2}(s-1)(s+5)}$

I'm seeking a partial-fraction expansion of $A=\frac{1}{(s+1)^{2}(s-1)(s+5)}.$ I was solve equation differential using Laplace transform, but I need use partial fraction of $A$.
0
votes
1answer
47 views

Definition of the infimum of a set in terms of a minimum.

I have the following intuition regarding the relationship between the infimum of a set: Let's define the set of accumulation points of a set called $S$ as $S'$, its closure:$\bar{S}=S \cup S' $, and ...
1
vote
1answer
28 views

Riemann Stieltjes Mean Value theorem - result

This is a well known result of Riemann Stieltjes integration: All the proofs I found use the fact that $f$ is bounded and apply one the Mean Value Theorem Riemann-Stieltjes Integrals (this one). I ...