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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Paradoxical question about infinite bump function.

Suppose that you have a bump function of the form $f\left( u \right)=e^{-\left( \frac{1}{1-\left( u \right)^{2}} \right)}$ that is continuous on $\{u$:$\\\ \mid u \mid < 1\}$ Now compose that ...
3
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1answer
16 views

Density of measurable sets in $\mathbb{R} $

Let $A$ be a Lebesgue measurable set in $\mathbb {R} $. We can classify the points in $\mathbb{R}$ as 3 disjoint subsets: density 0 points $A_1$, density 1 points $A_2$, otherwise $A_3$. By the ...
4
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1answer
17 views

Open set in a general metric space.

Let d define a metric on an infinite set M. Show that there exists an open set U such that U and its complement are infinite. (Infinite referring to cardinality in both instances) I know this is ...
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13 views

Some real analysis questions about first category

Let $\{x_n(t)\}_{n=1}^{\infty}$ be a set of continuous function from $[0,1] \rightarrow \mathbb{R}$, and $x(t)$ be the function such as $\lim_{n \to \infty} x_n(t) = x(t)$ (pointwise convergence). ...
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27 views

proof with 3 quantifiers?

Problems: ∃x ∈ S s.t. ∀y ∈ S, ∀z ∈ S, if z > y, then z ≥ x + y. ∀x ∈ S, ∃y ∈ S s.t. ∀z ∈ S, if z > y, then z ≥ x + y. What I have known about FOL: (∀ ∃) in calculus, find a function N=f(epsilon) or ...
3
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1answer
28 views

When is a continuous function piecewise monotone?

Given a continuous function $f:[a,b]\mapsto \mathbb{R}$, are there known additional conditions that ensure $f$ is piecewise monotone? Like this question, my motivation is to decompose the interval ...
4
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1answer
23 views

If one side of $\int f\ d\lambda = \int f\ d\mu - \int f\ d\nu$ exists, does the other one exist as well?

Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting ...
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29 views

How to write a “set is open” or “set is closed” in a pure symbolic way with quantifiers? (FOL)

How to write a "set is open" or "set is closed" or "a set is open" in a pure symbolic way with quantifiers? And how to use pure symbol to prove "E is open iff its complement is closed"? and that ...
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10 views

The question about the support of Fourier transform of $|f|^p$

Suppose $f$ is a smooth function with $\mathbb{supp}{(\mathcal{F}{f})} \subset B(0,1)$. In addition, assume $f$ is positive. We can observe that $|f|^2$ has some nice property : ...
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1answer
24 views

Help with a proof about limit point

$\{x_k\}$ is a sequence of real numbers and let $\Omega$ be the set of all distinct points in $\{x_k\}$. Prove that if $\Omega$ has a limit point $x$, then $\{x_k\}$ has a sub-sequence converging to ...
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1answer
24 views

Radon measure and a non-L1 function

This is a part of the exercise 7.17 in Folland's Real Analysis: Suppose $\mu$ is a positive Radon measure on a locally compact Hausdorff space $X $ with $\mu (X)=\infty. $ Show that there exists ...
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1answer
33 views

Principle of well ordering

Every non-empty set $A\subset\mathbb{N}$ have a smallest element, i.e. an element $n_0\in A$ such that $n_0\leq n$ $\forall n\in\mathbb{A}$ Proof: Let $I_n=\{p\in\mathbb{N};p\leq n\}$ the set ...
3
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2answers
30 views

Cantor Sets in perfect sets in the Real numbers

My thesis is related with the Cantor sets. I was reading a lot of papers, blogs, etc, in order to look for the mean properties of these sets. In one blog a read a proposition. ''Every perfect set ...
3
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2answers
177 views

If the derivative is zero on [a, b] so the function is constant - using Heine-Borel?

I know the proof using MVT but I was wondering if it can be proofed using Heine-Borel Lemma, that "Every open cover of close interval has a finite subcover". (without compactness, simple as that). ...
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0answers
21 views

Find the Lipschitz constant of a multi-variate Gaussian density function

I would like to find the Lipschitz constant of a multi-variate Gaussian density function: $$f_{\mathbf x}(x_1,\ldots,x_k) = \frac{1}{\sqrt{(2\pi)^{k}|\boldsymbol\Sigma|}} ...
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4answers
33 views

For a finite set in $\mathbb{R}$, the interior is empty and the closure and boundary are the set itself

How do I show explicitly that for a finite set in $\mathbb{R}$ the interior is empty and the closure and boundary are the set itself? For closure is simple: it is union of boundary and interior.But ...
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1answer
15 views

Consistency of differentiable functions on a closed subset.

I have a question about differentiable functions. Let $U \neq \emptyset$ be a open subset of $\mathbb{R}^{n}$ and $F$ be a closure of $U$. I want to define the space of infinitely differentiable on ...
2
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1answer
34 views

What is the meaning of $\lim_{\Delta(P) \to 0} F(P) = L$ for partitions

Let $[a,b]$ be an interval, and denote by $\mathcal P[a,b]$ the family of all partitions of $[a,b]$, i.e. sets $P = \{ a = x_0 < x_1 < \ldots < x_n = b \}$. For some $P \in \mathcal P[a,b]$ ...
2
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2answers
40 views

Does there exist a subsequence whose intersection has measure greater than $1/2$?

I ran across the following problem on this review guide. It is problem 1.25, though I've changed the wording slightly. The measure is implicitly Lebesgue measure. Let $E_n$ be a sequence of ...
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1answer
30 views

well defined mapping-function

I would like to know how to show an mapping or function is well defined i think in generale we use that : -$f$ is well defined mapping iff $( x\in E\implies f(x)\in F)$ in particular when mapping ...
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27 views

Question 10 from N.L Carothers' “Real Analysis”, Chapter 11.

Let $(x_{i})$ be a sequence of numbers in $(0,1)$ such that the $\lim_{n\to \infty}(1/n)\sum_{i=1}^{n}x_{i}^k$ exists for every k=0,1,2,.... Show that $\lim_{n\to \infty}(1/n)\sum_{i=1}^{n}f(x_{i})$ ...
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1answer
20 views

Does smoothness imply boundedness? Evans PDE chapter 2 Problem 18

In problem 18, enter link description here 1) I am having difficulty in extracting information in deciding the bounded for $g$ and $h$. In particular, to conclude $g$, $Dg$, $h$ $Dh$ are bounded by ...
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1answer
28 views

Unit radial vector field

Lee's book defines the unit radial vector field in normal coordinates as $$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$ and $r(x):=\sqrt{\sum_i (x^i)^2}$ Now this is a unit vector field iff ...
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2answers
33 views

Let $f(x) = [x], x \in [1,3]; \ \phi(x) = x , x \in [1,2]$ and $= 2x -2, x \in (2,3]$.show that $\int_1^3 f = \phi(3) - \phi (1)$

Let $f(x) = [x], x \in [1,3]; \ \phi(x) = x , x \in [1,2]$ and $= 2x -2, x \in (2,3]$. Then to show that $f$ is integrable and evaluating the value of $\int_1^3 f$. I have done upto this. But ...
4
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1answer
37 views

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma: If $x$ is irrational, there are infinitely many rational numbers $h/k$ with $k\gt 0$ ...
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0answers
23 views

Support of Radon measures

I am reading Folland's Real Analysis. The following is the exercise 7.2.b. Let $X$ be a locally compact Hausdorff space with a Radon measure $\mu$. Show $x\in\text{supp}(\mu)$ iff $\int f \text ...
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42 views

Convergence conditions

While studying for my exam in real analysis, I came up with the following problem. Given two sequences $\{a_n\}\to 0$ and $\{b_n\}\to\infty$. What should be the weakest condition so that $$\sum a_n ...
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4answers
74 views

$ \lim_{x \rightarrow \infty} e^{1/x} = a $ is not equivalent with $ \lim_{x \rightarrow \infty} a^x = e$?

I have problems understanding, why $$ \lim_{x \rightarrow \infty} e^{1/x} = a $$ is not equivalent with $$ \lim_{x \rightarrow \infty} a^x = e. $$ In the first case there is a solution $a=1$, and ...
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0answers
35 views

Concave optimization and corner solution

I have a optimization problem as follows: Assumptions: $f$ is an increasing and convex function on $R^+$ such that: $f(x): R^+\rightarrow R^+, \quad f(0)=0, \quad f'(x)\ge1,\quad f''(x)\ge 0 ...
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2answers
25 views

Minimum distance between closed sets.

I understand that if given a compact set K and a closed set C,that are disjoint, of a metric space then it follows that there is a minimum distance between them(You can prove this via a continuous ...
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2answers
51 views

Continuity in $\mathbb R$ results in continuity in $\mathbb R^2$; Proof?

During studying of proof of some other theorem, I faced with the claim (without proof): since $f(x,t)$ and $g(x,t)$ are continuous functions [$f,g:\mathbb R^2 \rightarrow \mathbb R$] thus the ...
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1answer
19 views

Question about proof extending measure to complete measure

I am looking through a proof in Folland, for Theorem 1.9, which states: Suppose that $(X, M, \mu)$ is a measure space. Let $N = \{N' \in M : \mu(N') = 0\}$ and $M' = \{E \cup F : E \in M' \text{ and ...
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Proving inner measure equal to outer measure if a set is measurable

I'm doing the problem 19 in Real Analysis of Folland like below: Let $\mu^*$ be an outer measure on $X$ induced from a finite premeasure $\mu_0$. If $E \subset X$, define the inner measure of ...
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2answers
28 views

Limit of Uniformly convegent sequence of real valued functions:

I was attempting this qualifying problem. $\{f_n\}_{n=1}^{\infty}$ is a sequence of real valued functions on $\mathbb{R}$. If $ f_n $ converges to $f$ uniformly then $f$ must be continuous. I am ...
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1answer
84 views

Every function $f: \mathbb{N} \to \mathbb{R}$ is continuous?

This is a question that came up as a true false question in my textbook, and I was wondering what you thought of my reasoning. I claim that even though a graph of such a function doesn't look ...
1
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1answer
48 views

let $\phi (x) =\lim_{n \to \infty} \frac{x^n +2}{x^n +1}$; and $f(x) = \int_0^x \phi(t)dt$. Then $f$ is not differentiable at $1$.

For $x \geq 0$, let $\phi (x) = \lim_{n \to \infty} \frac{x^n +2}{x^n +1}$; and $f(x) = \int_0^x \phi(t)dt$. Then $f$ is continuous at $1$ but not differentiable at $1$. First we calculate $\phi (x) ...
1
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1answer
76 views

How do I prove that $ax+by+cz=d$ has infinitely many solutions on $S^2$?

Let $a,b,c,d$ be reals such that $a^2+b^2+c^2>d^2$. How do I prove that $\{(x,y,z)\in S^2: ax+by+cz=d\}$ is infinite? This is geometrically trivial, but I'm stuck at proving it rigorously..
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1answer
32 views

Construct a special measurable functions

Suppose $A$ is a measurable subset of $[0,1]$, show that there exists a measurable function $f(x)$ on the interval $[0,m(A)/2]$ such that $$m([x,f(x)]\cap A)=\frac{1}{2}m(A)$$ for all $x$. I know ...
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1answer
23 views

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Let $\Omega\subseteq\mathbb{R}^n$ be a domain with a smooth boundary $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $p>1$ such that $$p<\begin{cases}\infty&\text{, if ...
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38 views

limit of a complex expression

Suppose that $$R(r)=\left[(a_0-r^2a_2)e^{ir\tau}+(b_0-r^2b_2)\right]^{\frac{1}{r}}\;,$$ where $a_0,a_2,b_0,b_2\in\mathbb{R}$ and $\tau>0$. What is $\displaystyle\lim_{r\rightarrow\infty}R(r)$? I ...
3
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2answers
97 views

Computing $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$.

Compute $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$ with a precision (Accuracy? Error? What is the formal expression?) of 0.01. Attempt: First of all: $\ln(x+1)=\sum_{k=1}^{\infty}{(-1)^{k-1}x^k\over ...
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1answer
19 views

Prove that $\sigma$-algebras $A_1,\ldots,A_n$ are independent if and only if $A_i$ is independent of each $A_1,\ldots,A_{i-1}$, for all $i=2,\ldots,n$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathcal{A}_1,\ldots,\mathcal{A}_n\subseteq 2^\Omega$ be $\sigma$-algebras. How can we show, that ...
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60 views

Show that the cardinality of the preimage is measurable

Suppose $f$ is continuous in $\mathbb R$ such that $|\{x:f(x)=y\}|<\infty$ for any $y\in \mathbb R$(where $|\cdots|$ means cardinality). Show that $$ g(y)=|\{x:f(x)=y\}| $$ is a measurable ...
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45 views

Two ODEs, why is one solution the solution of the other?

Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t. $$u'(t) = F(t,u(t))$$ $$u(0) = u_0$$ given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ Caratheodory, and we know that if it has a solution, it ...
6
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1answer
46 views

Finding a better upper bound for an integral of a product of $n$ terms

So I'm trying to find and upper bound for the integral $$ \int\limits_{a}^b \! (x-x_1)^2 \cdots (x-x_n)^2\, \mathrm{d}x, $$ where $x_i \in [a,b], \enspace \forall i=1,\dots ,n.$ I've tried ...
0
votes
1answer
21 views

$L^1 ([0,1])$, bouned linear functional, absolute continuous function

I am studying for an Analysis prelim and was wondering if someone could perhaps either validate or invalidate my proof for the following problem: "Let $L^1 ([0,1])$ be the space of Lebesgue ...
1
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0answers
36 views

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$?

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$? I have gone through all the identities relating Fourier series and I can't seem to understand why. In this question, the ...
0
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4answers
46 views

Must a continuous function on $\mathbb R$ with only rational values be constant? [duplicate]

As I'm preparing for my exam I have to solve the following question: Determine if the following is correct: Let $f$ be a continuous function is $\Bbb R$. If $f$ recieves only rational values, ...
0
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0answers
30 views

Induction on derivatives

I have troubles understanding this induction proof: Let $$g(x) = \vert x \vert^{2k+1}$$ Show by induction: $$\frac{\partial ^N g(x)}{\partial x_{i_1} \dots \partial x_{i_N})} = cx_1n \dots x_iN \vert ...
4
votes
1answer
58 views

Condition for Continuity (two variable)

I came across the following question while studying for quals. This one is from a previous qualifier. I have a few ideas (which I'll mention below), but am stuck on how to complete the problem. Any ...