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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11 views

“Continuity” of minimum of a function

Following is a question I encounter while reading a topic on 'large deviations'. I abstract the problem here. I don't know whether the title is apt! Let $\mathbb{A}=\{a_1,\dots,a_d\}$ be a finite ...
2
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0answers
31 views

Is this a correct proof of the inverse function theorem?

This question concerns the proof of the inverse function theorem found in Walter Rudin's Principles of Mathematical Analysis (3rd ed.). I have found an alternate proof of one part of the theorem but I ...
2
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1answer
47 views

Can this integral diverges?

Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be a continuous non-increasing bounded function, and suppose $$\limsup_{x\to +\infty}\frac{f(x)}{f(2x)} = +\infty. $$ Can $\int_0^{+\infty} f(x)dx$ diverges? ...
0
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1answer
14 views

Prove $\sup_x|u_x(x,t)|\le Ct^{-\frac{3}{4}}\|f\|_2$ for all $t>0$.

Let $u$ be a bounded solution to the heat equation $u_t-u_{xx}=0$ in $-\infty<x<\infty,t>0$ with $u(x,0)=f(x)$ with $f\in L^2(\Bbb R)$. Prove that there is a constant $C>0$, independent ...
2
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2answers
18 views

A proof of a known identity using Fourier series

The exercise Let $f$ be a $2\pi$ periodical function defined as $f(x)=\cos ax, \; |x|\leq \pi, \; a \notin \mathbb{Z}$. Expand $f$ in a Fourier series and prove that: $$\pi \cot \pi a = ...
3
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1answer
31 views

Haar measure - a problem from Folland

I was presented with this question from Folland's real analysis second edition involving Haar measures. It is problem 3 of chapter 11 page 347, which reads as follows: Let G be a locally compact ...
2
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0answers
22 views

Natural extension of a discontinuous function

Let $u : \mathbb{R} \to \mathbb{R}$ be the right continuous version of the Heaviside step function. What does the natural extension $u^*$ of $u$ to the set $\mathbb{R}^*$ of the hyperreals look like? ...
0
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1answer
28 views

Showing that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^\infty$ [on hold]

I want to show that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^\infty$, however I am not to sure how to go about doing so. We have, $$d_\infty(x,y)=\sup_i|x_i-y_i|$$ and $\mathscr ...
0
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1answer
30 views

What can be said about the map?

$f\colon \mathbb R^{2}\rightarrow \mathbb R$ is a continuous map that assumes $0$ for only finitely many points. Then which one is true A. either $f(x)\le 0$ for all $x$ or $f(x)\ge ...
1
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1answer
33 views

Good reference on higher dimensional derivatives?

I've spent several months now periodically scouring the internet for a comprehensive overview of an introduction to higher dimensional derivatives. I've already read baby Rudin's section on the ...
0
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1answer
24 views

Proof that outer measure of interval equals length, why use Heine-Borel?

Define by $$ m^*(A) := \inf\left\{ \sum_i |I| : A \subseteq \bigcup_i I_i \right\} $$ the outer measure of some set $A \subseteq \mathbb R$. Then we have $m^*(I) = |I|$ for each interval (open, ...
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3answers
446 views

Is it mathematically wrong to prove the Intermediate Value Theorem informally?

I have been looking at various proofs of the IVT, and, perhaps the simplest I have encountered makes use of the Completeness Axiom for real numbers and Bolzano's Theorem, which, honestly, I find a bit ...
0
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1answer
16 views

Reiterating the piecewise-and-uniform-limit operation

Probably a hopeless question, but: Let $C$ be the class of constant functions $f$: $[a, b]\longrightarrow\mathbb{R}$. Let $\mathcal{U}(\mathcal{P}(C))$ denote the class of uniform limits of piecewise ...
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2answers
24 views

Problem with understanding the application of the Intermediate Value Theorem in the proof of the Mean Value Theorem for Integrals

I am struggling to understand the last parts of this proof because I know that the IVT states that on the interval $[a,b]$ of $f$, where it is continuous, there exists a value $L$ between $f(a)$ and ...
-1
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1answer
44 views

The existence of a smooth functions taking values $0$ and $1$ on two given closed sets [on hold]

In theorem 5.1 on page 39 Boothby's book(An introduction to Differentiable Manifolds By William M. Boothby), he prove that: Let $F\subset \mathbb R^n$ be a closed set and $K\subset \mathbb R^n$ ...
2
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2answers
30 views

$\sup_{x\in B_r(z)}|\nabla u(x)|\geq cr$ for $\Delta u=1$?

Let $U\subset\mathbb R^n$ be open and bounded. Let $\Delta u=1$ in $U$. Can you follow $\sup_{x\in B_r(z)}|\nabla u(x)|\geq cr$ provided $B_r(z)\subset\subset U$? I thought about using the ...
0
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2answers
27 views

Metric spaces, manipulating the absolute value function.

I have the following problem involving the set $Y$ of infinite sequences that absolutely converge such that, $$\sum_{i=0}^\infty x_i^2 \lt\infty$$ where $x_i$ is the $i$-th term in the infinite ...
1
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1answer
43 views

Function $f \geq 0$ a.e.

Suppose $f : \mathbb{R}^d \to \mathbb{R} $ is integrable (with respect to measure $\mu$) and for every measurable set $E \subset \mathbb{R}$ we have $\int_{E} f d \mu \geq 0$. Prove that $f \geq ...
1
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0answers
26 views

Surface integral on unit circle

Let $S$ be the unit sphere in $\mathbb{R}^3$ and write $F(x)=\nabla V(x)$ where $V(x)=1/|x|$ Evaluate $$\iint_S F\cdot n dS$$ Without using divergence theorem, we can evaluate it straightforwardly, ...
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0answers
20 views

Show that a collection of finite unions of sets of the form $(a,b]\cap \mathbb{Q}$ is an algebra

The following is a question from Folland's Real Analysis: Modern Techniques and their Applications. (Question 23 page 32) Let $\mathcal{A}$ be the collection of finite unions of sets of the form ...
2
votes
1answer
44 views

“Mean value like” problem.

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be differentiable, take $a<a'<b<b'$. Prove that there exists $c<c'$ such that $$\frac{f(b)-f(a)}{b-a}=f'(c) \quad and \quad ...
1
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1answer
20 views

Improper integral parametrised in complex variable: when is it holomorphic?

Suppose we are considering the following integral: $$ I(s) = \int_1^\infty t^{s-1}e^{-t\lambda}\;dt $$ where $s \in \mathbb{C}$ and $\lambda > 0$ is a fixed constant. I would like to know when this ...
0
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3answers
53 views

Find x for which $\sum [(n^3+1)^\frac{1}{3}-n]x^n$ converges.

For what values of $x$ the infinite series $\sum [(n^3+1)^\frac{1}{3}-n]x^n$ converges? Please help me out. Using ratio test $\large{\lim_{n\to ...
3
votes
0answers
28 views

Can we find real values of $x$ such that $\lim_{n\to \infty}c^n\prod_{k= 1}^n| a_k x-1|=0$?

Let $\{a_n\}$ be a sequence such that the sequence of Cesaro means is convergent i.e. $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n a_k$$ exists. Let $c>1$ be a given number and I am trying to find ...
0
votes
2answers
66 views

Is it possible to find $a$, $b$ such that $(a,b] \cap \mathbb{Q} = \emptyset$?

Is it possible to find $a$, $b$ $\in \mathbb{R}$ satisfying $- \infty \leq a < b \leq \infty$ such that $(a,b] \cap \mathbb{Q} = \emptyset$? I need this result for a proof I am doing...
-1
votes
1answer
38 views

How to calculate $lim_{n\rightarrow \infty} \dfrac{n+\sin 2n}{\sqrt{n}-\cos n }$ [on hold]

I have to solve this problem. $$lim_{n\rightarrow \infty} \dfrac{n+\sin 2n}{\sqrt{n}-\cos n }$$
0
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1answer
33 views

Exercise 43 chapter 2 in Real Analysis of Folland

I got stuck on this problem and couldn't find any clue to solve it. Can anyone give me some hint or give me some solution for it. I really appreciate! Suppose that $\mu(X) < \infty$ and ...
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0answers
13 views

Conditional density of degenerate multivariate normal

Let $X_{12},X_{13},X_{14},X_{23},X_{24},X_{34}$ be identically normal $N(\mu,\sigma^2)$ such that every linear combination among $X_{ij}$'s is also normal, $corr(X_{ij},X_{rs})=\rho$ if ...
5
votes
2answers
100 views

Is $(a,a]=\{\emptyset\}$?

Let $a \in \mathbb{R}$, and consider the half open interval $(a,a]$. Is it correct to write this half open interval as $(a,a]=\{\emptyset \}$? Or $(a,a]=\{a \}$?
-1
votes
2answers
22 views

Proof structure validity: assume (a), (b), show (c). Then Permute.

I am given a collection of sets $\mathcal{E}$ and am trying to prove it is an elementary family. To show $\mathcal{E}$ is an elementary family I must show that it satisfies the following properties: ...
0
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2answers
43 views

Real analysis using some key constraints

Let $\alpha>1$ and $M \geq 0$. Suppose $f:\mathbb{R}→\mathbb{R}$ satisfies $$|f(x)-f(y)| \leq M|x-y|^\alpha$$ for all $x,y\in\mathbb R$. Prove that $f$ is a constant function. I tried taking ...
1
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1answer
39 views

Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in ...
1
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0answers
20 views

Conditions on $f$ to have $ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{f(x)}{(x-y)^2 (y-z)} dz dy dx $ finite?

Suppose that $f$ is a $\mathcal{C}^\infty$ function. $$ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{f(x)}{(x-y)^2 (y-z)} dz dy dx $$ Which are the conditions on $f$ that makes this integral finite ? ...
2
votes
2answers
62 views

Show $\int_E {(f_1 + f_2)d\mu } = \int_E {f_1 d\mu } + \int_E {f_2 d\mu } $

In my textbook, given a measure space $(\Omega,F,\mu)$, the integration for a non-negative $F$ measurable function $f$ on $E$ is defined as $$\int_E f\ \mathsf d\mu = \sup_{0 \le h \le f} I_E\left( h ...
0
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0answers
29 views

Why do we need a Borel function in order to use this lemma?

Im trying to understand a proof for differentiably a.e for functions $F$ given by $$F(x)= \int_{-\infty}^{x}f\ \mathsf dt$$ for $f$ Lebesgue measurable and $L^{1}$. He defines a finite Borel measure ...
0
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1answer
15 views

The relationship between function space embeddings and their respective inequalities

Let $L^{p,\infty}$ be the weak $L^p$ space consisting of measurable functions $f$ satisfying \begin{equation*} ||f||_{p,\infty}:=\sup_{\rho}\rho\lambda (|f|>\rho)^{\frac{1}{p}}<\infty . ...
0
votes
1answer
10 views

Approximation by $\mbox{Im }(t-z)^{-1}$ with $\mbox{Im } z > \epsilon$

It is a standard fact of harmonic analysis that the span of the functions $$g_z(t) = \mbox{Im } (t-z)^{-1},$$ ranging over all $z \in \mathbb{C}$ with $\mbox{Im } z > 0$, is dense in ...
0
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0answers
55 views

If $f'(ax) = \frac{b}{a}$ then $\frac{\textrm{d} x}{\textrm{d} a} > 0$ where$f, f' > 0$ and $f'' < 0$?

I'd appreciate any help I could get for this question: Define $f:[0, +\infty) \rightarrow R^+$ with $f(0) = 0$ and $f(\infty) = \infty$, and such that $f' > 0$ and $f'' < 0$. I have the ...
1
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0answers
21 views

$e\cdot(m(m-1)+1)\cdot k\cdot ( 1-\frac{1}{k})^m\leq 1$

I have to show that $e\cdot(m(m-1)+1)\cdot k\cdot ( 1-\frac{1}{k})^m\leq 1$ holds for all positive integers k and m whenever $m>4\cdot k\cdot log(k)$. I replaced $(1-\frac{1}{k})^m$ with ...
1
vote
0answers
13 views

Show measures are aboslutely continuous based on relationship between integrals of certain functions

So I have the following question. Suppose $\mu$ is a regular Borel measure on $[0,1]$ and $\{f_n\}\subseteq L^1[0,1]$ s.t. $|f_n(x)|\leq |f(x)| \ [m]$ a.e. and $f\in L^1[0,1].$ Given ...
3
votes
1answer
43 views

Injectivity of transformation

Is transformation $g:x=(x_1,x_2,x_3)\mapsto \frac{x}{\|x\|}$ injective? What if $x_1=1$?
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1answer
47 views

An example of discontinuous function on $\mathbb{R}$ [duplicate]

Is there any example of a function which is discontinuous on $\mathbb{R}$
2
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0answers
24 views

Least expensive way to “walk” with a convex potential

Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= (r_i)_{i=1}^{N}$ such that $\sum\limits^N_{i=1} r_i ...
0
votes
0answers
12 views

A pde that cannot solves by Lax-Milgram theorem

Consider the following pde: $-u''(x)+au'(x)+bu(x)=f(x) \qquad\text{in}\; (0,1)\\$ $u'(0)=\alpha\\$ $u'(1)+u(1)=\beta$ How could I prove that it has a nontrivial solution? The bilinear form ...
3
votes
1answer
25 views

Manipulating the maximum function, metric spaces.

I am trying to show that the supremum metric, $d_{\infty}$, is indeed a metric on $\mathbb R^2$. I have shown that the first two properties of a metric space hold, but am having trouble showing the ...
1
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0answers
33 views

Proof of additivity of domain for definite integrals

I would like to prove the following theorem: Theorem If $c \in (a,b)$ and $f$ is integrable on $[a,c]$ and $[c,b]$, then $f$ is integrable on $[a,b]$ and $$\int_{a}^{b}f = \int_{a}^{c}f + ...
0
votes
3answers
42 views

The derivative function is not continuous

(Sorry about the bad title, couldn't think of a way to word it concisely.) Let $C[0, 1]$ be the metric space whose points are all continuous functions from $[0, 1] \rightarrow \mathbb{R}$ with the ...
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votes
0answers
36 views

Pugh real mathematical analysis - chapter 1 - exercise 14 [on hold]

I absolutely have no idea for solving the problem... any help would be appreciated!
8
votes
1answer
51 views

Show that $\lim_{t\to \infty}u(x,t)=\frac{A+B}{2}$, for each $x\in\Bbb R$.

Let $u(x,t)$ be a $C^2$ bounded solution of $$u_t(x,t)-u_{xx}(x,t)=0,x\in \Bbb R, u(x,0)=f(x)$$ where $f\in C(\Bbb R)$ satisfies: $\lim_{x\to+\infty}f(x)=A,\lim_{x\to-\infty}f(x)=B$. Show that ...
5
votes
1answer
112 views

Cauchy-Schwarz Inequality without using $\langle a x,y\rangle=a\langle x,y\rangle$

Let $V$ be a vector space and define a function $\langle .,.\rangle:V\times V\to\mathbb{C}$ such that $$\begin{align} & \langle x,y\rangle=\overline{\langle y,x\rangle }\,\,\,\forall x,y\in V\\ ...