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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1answer
13 views

A real integral (may be requires contour integration)?

The integral I have in mind is $$\int^\infty_0 x^{r}(x + \lambda)^{-1}dx$$ where $r \in (-1, 0)$, and $\lambda$ is a non-negative constant. I apologize if this is really easy and I am missing some ...
-1
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1answer
16 views

If $u_k$ converges uniformly on $\partial \Omega$, does it converge uniformly on $\Omega$?

Let $u_k$ be continuous on $\overline \Omega$ and harmonic in $\Omega$. Suppose $u_k$ converges uniformly on $\partial \Omega$. Can we conclude that $u_k$ converges uniformly on $\Omega$?
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1answer
23 views

$L^1([0, 1]) \subset C([0, 1])^*$

Basically my question is: how can I prove that $L^1([0, 1]) \subset C([0, 1])^*$, where $C([0, 1])$ represents all continuous functions on $[0, 1]$, and the superscript $^*$ means the dual space. ...
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2answers
47 views

Find functions $f$ and $\alpha$ such that $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist?

Find functions $f$ and $\alpha$ such that the improper Riemann-Stieltjes integral $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist? I'm really not sure how to start ...
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0answers
24 views

Convergence of the integral of a product of functions.

Let $\phi:\mathbb{R^n}\to\mathbb{R}$ be a Lebesgue-measurable function, with the property that for every $n$-dimensional cube $Q$ in $\mathbb{R^n}$, we have $$ \left|\int_{Q}\phi(x)dx ...
1
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0answers
16 views

Trigonometric polynomial

Let $f \in L^2$ be compactly supported, then I want to show that $$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$ ...
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0answers
23 views

Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
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4answers
26 views

Proof $log_{r} a = log_r s \cdot log_s a $

Do you know any proof of this logarithms property: $log_{r} a = log_r s \cdot log_s a $
0
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2answers
19 views

A probability theory question

let X be a rondom variable and coonsider a non-negative function $g: \Bbb R \to \Bbb R^+$ Please help me sshowing this following statement; for $r\in \Bbb R^+ $, $$P(g(X)\gt r) ...
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0answers
23 views

On the rearrangement of an infinite series of real numbers. [duplicate]

A chapter of a text book ended with. If we rearrange infinitely terms of a series that converges only conditionally, we may get results that are far different from the original series ...
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0answers
26 views

Folland, “Real Analysis”, Chapter 5.3, Exercise 36.

Folland, "Real Analysis", Chapter 5.3, Exercise 36: Let $\mathcal{X}$ be a separable Banach space and let $\mu$ be counting measure on $\mathbf{N}$. Suppose that $\left\{x_n\right\}_1^\infty$ ...
2
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1answer
32 views

How to pick decimal expansion in the proof that $(0,1)$ uncountable

Prove $(0,1)$ is uncountable. Suppose $(0,1)$ were countable. List $(0,1)$ as: $x_1=0.a_{11}a_{12}\dots$ $x_2=0.a_{21}a_{22}\dots$ and so on, where $a_{ij}$ are integers from $0$ to $9$. ...
1
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1answer
25 views

curves and integral

Find the area between these curves. $$y=\dfrac{3}{2x+1},\qquad y=3x-2;\qquad x=2\quad \text{et} \quad y=0 $$ indeed, I calculate the integral of the blue function between $1$ and $2$. Then, I ...
2
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0answers
18 views

Prove that disk algebra is isomorphic to the closure of $\mathbb{C}(z)$ in $C(\mathbb{T})$.

Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ be the open unit disk and $\mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}$ its boundary. We will naturally write $\bar{D}$ for its closure $\{ z \in ...
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0answers
15 views

Real analysis-proving limit existence for an evaluated function

I don't understand how to define delta for this problem
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1answer
31 views

proving a statement based on probability theory [on hold]

Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$
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2answers
34 views

A question related to measura space

Let a real value $X$ be a random variable and consider $\int_{\Omega}|X|dP \lt \infty $. I need to show that \begin{equation*} nP(|X|\gt n)\to_{n\to \infty} 0. \end{equation*} please help me ...
3
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1answer
28 views

Hessian-Matrix positive definite $\iff$ $a$ local minimum?

It is commonly known that if $f$ is twice differentiable, $\nabla f(a) = 0$ and $H_f(a)$ positive definite, $a$ is a local minimum. So, in short: $H_f(a)$ positive definite $ \implies $ $a$ local ...
3
votes
1answer
43 views

length of the curve $y=x^n$ in the unit square

Let $l_n$ be the length of the curve $y=x^n$ in $[0,1]\times[0,1]$. Then obviously $\lim_{n\to\infty}l_n = 2$. What about $\lim_{n\to\infty}(n(2-l_n))$ ? The formula $l_n = ...
2
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1answer
67 views

Study the following integral: $\int_0^\infty \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x … (\ln^{(k)} x)^s }$

How do I calculate for which values of $s$ the following integral converges? $$\int\limits_{0}^{\infty} \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} ...
5
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2answers
80 views

Limit of $\frac{f\left(x+g\left(x\right)\right)-f\left(g\left(x\right)\right)}{x}$ as $x\to 0$

I'm trying to answer the following question: Let $f$ be continuously differentiable in all of $\mathbb{R}$ and let $g:\mathbb{R}\to\mathbb{R}$ be a function satisfying ...
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2answers
30 views

Using the same limit for a second derivative

I've been trying to answer the same question answered here: Second derivative "formula derivation" And I'm stuck in a step that is not addressed both in the answer and in the comments of ...
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0answers
16 views

Approximating roots

Given $n,r\in\Bbb N$, assume $a=n^\frac{1}r$. Assume that $a_d$ is $a$ truncated to $d$ digits ($d$ is total digits both before and after decimal Eg: truncating $412.243$ to $2$ digits is $410.000$ ...
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0answers
12 views

diffeomorphism on intervals on R

I came across a line in a proof which involved a diffeomorphism $f:I_1 \rightarrow I_2$ (with $f$ a homeomorphism, $f,f^{-1}\in C^{\infty}$) mapping open intervals in R, which claimed that ...
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0answers
17 views

Second derivative and Hessian-Matrix

Suppose $f$ is twice differentiable. Why is $Df(a)[v] + \frac12 D^2f(a)[v,a] = \frac 12 \langle H_f(a) v, v \rangle $ ? $Df(a)[v]$ is the multidimensional derivative of $f$ at point $a$ in ...
1
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4answers
68 views

Calculate the limit as $x\to0$

I need to calculate the limits as $x$ tends to $0$ For the first one, I get that the limit is zero, by splitting it up into $x^3(\sin(1/x))$ and $x^3(\sin^2(x))$ and using the sandwich theorem on ...
2
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1answer
18 views

Fourier transform of $L^1$ function square summable?

It is known that for a $L^1$ function $f: \mathbb{R} \rightarrow \mathbb{C}$ the Fourier transform vanishes at infinity and is continuous. Does this even mean that $(\hat{f}(n))_{n \in \mathbb{Z}}$ is ...
6
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1answer
47 views

which functions can be obtained as a composition of a continuous function with itself? [duplicate]

let $f(x)=x^2$, then $f(f(x))=x^4$, so $x^4$ is a continuous function from $\Bbb R$ to $\Bbb R$ which can be obtained as $f\circ f$ for a continuous $f\colon \Bbb R\to \Bbb R$. general example: for ...
0
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1answer
28 views

metric space: equivalence of several mertric.

I have two questions: Q1) Are all metric on a metric space are equivalent ? Q2) If not: Let $d_1,d_2$ two metric on $X$. If something has a property with a $d_1$ will it hold for $d_2$ too ? For ...
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2answers
166 views

Proving the equation $C=2x/(1+x^2)^2$ has two positive solutions

I want to show the equation $C=\frac{2x}{(1+x^2)^2}$ has two solutions where $C$ is a constant such that $0<C<\frac{1}{2}$. My method was to say at $x=1$, $RHS=\frac{1}{2}>C$ and I'm sure ...
0
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1answer
20 views

Map between two metric spaces and their limit

I have to proof the following: Let $f: V \to W$ be a map between two metric spaces. Proof that $f(a)$ with $a \in V$ is continuous if and only if, for every converging sequence $x_n$ in V with limit ...
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3answers
57 views

Prove that $f : [a,b] \rightarrow \mathbb{R}$ is a bijection from $[a, b]$ to $[f(a), f(b)]$

I'm a 1st year mathematics student, and in my analysis class I'm having trouble with proving the following: Let $a < b \in \mathbb{R}$, and let $f : [a,b] \rightarrow \mathbb{R}$ be a continuous ...
1
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1answer
17 views

if one of the sets A and B is compact then d(A,B)>0.

Let $A$ and $B$ be two nonempty disjoint subsets of $\mathbb{R}^{n}$. Put $d(A,B)=inf\left \{ ||a-b||:a\in A, b\in B \right \}$. a) Show that if one of the sets $A$ and $B$ is compact then ...
2
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1answer
27 views

Let $\mathcal R$ be a $\sigma$-ring, then: $\{E\subset X; E\cap F \in \mathcal R\text{ for every } F \in \mathcal R\}$ is a $\sigma$-algebra.

I'm trying to Solve the following question: Let $X$ be a non empty set and $\mathcal R$ be a $\sigma$-ring from subset of $X$. Prove that: $$S=\{E\subset X; E\cap F \in \mathcal R\text{ for ...
0
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1answer
13 views

Finding the upper boundary of a sequence to find it's limit

I have to determine if the following sequences converge and if they converge I have to determine to what they converge and proof this. $$ a_n = 2^{-n} \\ b_n = \frac{n^2}{n^3 -10} \\ c_n = 1 ...
0
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2answers
19 views

A question involving continuity with respect to the product topology

Let H be a nonempty set, $\cdot$ a binary operation on H, $\Gamma$ a topology on H and $$\varphi : H \times H \to H, \;\; \varphi(x, y) = x y, \;\; \forall x, y \in H$$ continuous with respect to the ...
0
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0answers
10 views

Infinite-dimensional Rieman Integral and Monte Carlo method

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(\xi_k)_{k \in N}$ be a sequence of independent uniformly distributed on the interval $[0,1]$ real valued random variables such that $0 \le ...
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1answer
19 views

Check differentiablity of $f$ [on hold]

Consider a function \begin{equation*} f(x)=|\cos x|+|\sin (2-x)|. \end{equation*} At which of the following points f is not differentiable? a)$\{(2n+1)\frac\pi2:n\in \Bbb Z\}$ b)$\{n\pi:n\in \Bbb ...
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1answer
20 views

Determine whether the limit exists and justify answer

Usually for these types of questions, I use sequences of functions to show that the limit does not exist, but I don't think I can do this here? I feel like the limit should be zero, but I don't know ...
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1answer
42 views

Existence of divergent series $\sum_{n=1}^ \infty a_n$ of real numbers whose partial sums are bounded and $\lim (na_n)=0$ [on hold]

Does there exist a sequence $(a_n)$ of real numbers such that $\lim_{n \to \infty} (na_n)=0$ , the partial sums of $\sum_{n=1}^ \infty a_n$ are bounded , but $\sum_{n=1}^ \infty a_n$ is divergent ? ...
0
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0answers
16 views

Nonnegative harmonic functions

Suppose $U \in \mathbb{R}^n$ is an open domain, and $u\in C^2(U) \cap C(\bar{U})$ such that $\Delta u = 0$ in $U$. I'm working on a couple of problems pertaining to the mean value formula/harmonic ...
4
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1answer
74 views

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y \in \mathbb{R}^n$

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying \begin{equation*} f(x+y)+f(x-y)=2f(x)+2f(y)~\forall x,y \in \mathbb{R}^n. \end{equation*} My attempt: I manage to show ...
1
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1answer
20 views

Characteristic Function in the subset E

Let $E \subseteq \mathbb R$. Then the characteristic function $\chi_{E}:\mathbb R \to \mathbb R$ is continuous if and only if a) $E$ is closed. b) $E$ is Open. c) $E$ is both Open and Closed. d) ...
1
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1answer
28 views

What is the graphical representation of Lipschitz continuity?

I know the graphical represntation for Continuity, Uniform continuity,Absolutely continuity, but I want to know the graphical representation of Lipschitz continuity.
4
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1answer
24 views

Help understanding an inequality on Rudin's construction of the Lebesgue measure

I am having trouble understanding an inequality in Theorem 2.20 from "Real and Complex Analysis." Rudin states that if $f\in\operatorname{C}_c(\mathbb{R}^k)$ , $f$ is real, $W$ is an open k-cell ...
2
votes
1answer
21 views

Counterexample to the double integral computational theorem when the double integral existence assumption is dropped?

To make things simple, consider the simplest case of the double integral computational theorem. Throughout any phrase involving integrability is in the Riemannian sense. Let $[a, b], [c, d] \subset ...
0
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1answer
21 views

Orthogonal set of a set in Hilbert space

This is an exercise in the Folland Real Analysis p.177. I first thought it is an easy one, but it turns out to be a lot trickier..... I have no idea how to deal with the so-called "double ...
0
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0answers
16 views

Is it possible to modify norm of Sobolev space suitable for ill-posed problems.

I have trying to pose my problem mathematically for quite some time now. I am not sure even if I am close to defining properly. Would anyone please help: I am interested to study ill-posed problem of ...
2
votes
1answer
81 views

Calculate $\lim_{n \to \infty}(\sin nx)^\frac{1}{n}$

I know that $$(n)^\frac{1}{n} \to 1$$ and $(a)^\frac{1}{n} \to 1$ (with $a \in \mathbb{R+}$). However, I was wondering what can be said about $$\lim_{n \to \infty }(\sin nx)^\frac{1}{n}$$ and, more ...
1
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1answer
23 views

Is the function $f(x)=x^2$ absolutely continuous on the real line?

In Wiki (Lipschitz), it says: A Lipschitz function $g : \mathbb{R}\to \mathbb{R}$ is absolutely continuous. According to the definition of absolute continuity, I am confused about an simple ...