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, integration through the fundamental theorem of calculus.

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On the smallest sigma algebra of a set

Let $X$ be any set. Say $\mathcal{E} \in 2^X $. Then there exists a unique smallest $\sigma-$algebra containing $\mathcal{E}$ Attempt: Put $$ \mathcal{C} = \{ \mathcal{F} : \mathcal{F} \; \text{is a ...
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35 views

Euler and the factorial function

I recently purchased H. M. Edwards' book entitled The Riemann Zeta Function. In the early pages of the volume, concerning the factorial function $\Gamma$, Edwards notes that "Euler observed that ...
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51 views

Show that $f(x)$ is uniform continuity in $(0,1]$

Suppose that $f(x)$ is a continuously differentiable function in $(0,1]$,and $\lim\limits_{x\rightarrow0^{+}}\sqrt{x}f(x)$ exists. Show that $f(x)$ is uniformly continuous on $(0,1]$.
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Show $u\in H^1(B(0;1/2))$ is holder continuous, where $u$ is a weak solution to $-\Delta u+cu=f$ for some $c\in L^q$ for some $3/2<q<2,$.

If $u\in H^1(B)$, $B=\lbrace x\in\mathbb{R}^3, |x|<1/2\rbrace$ is a weak solution to $$-\Delta u+cu=f$$ for some $c\in L^q$ for some $3/2<q<2,$ and $f\in C^\infty$, then show $u$ is holder ...
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35 views

Sequence is bounded <=> lies in a compact subset?

In a book I'm reading, there's some theorems that prove some stuff about sequences that lie in compact subsets of some set $S$. Then, in a later theorem, we're asked to assume boundedness about some ...
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17 views

For which values of lambda is the set of line integrals bounded above?

Let P = {(x,y,z) $\in$ $R^3$ | 0$\le$ z$\le$1, 1$\le$$x^{2}$+$y^2$$\le$4}. For $\lambda$$\in$R, consider the vector field $$F_\lambda(x,y,z) = (2x+ \lambda y,-\lambda x+2y,2z) $$ in P. For which ...
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113 views

Is $\frac{1}{e^\gamma\log x} \prod\limits_{p < x,p\,\text{prime}} \frac{p}{p-1}<1+ \prod\limits_{p<x,p\,\text{prime}}\frac{1}{p^{n+1}-1}?$

Let $n$ be an initially arbitrarily large variable, but always decreasing (and more specifically non-increasing) to exactly $1$ when $p$ is the largest prime in the product. Then, denoting with ...
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1answer
12 views

Weighted Norm Minimization

I have a minimization problem of the form $min (w_1 \|x\|+w_2\|y\|)$ subject to constraints $A_1x=b_1$, $A_2y=b_2$, $0< l_1 \leq x \leq u_1$ and $0< l_2 \leq y \leq u_2$ where ...
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44 views

Measure Theory Problems [duplicate]

Let $\mathcal{E}$ be an arbitrary collection of subsets of a set $X$, let $A$ be a nonempty subset of $X$, and let $$ \mathcal{E}\cap A:=\{E\cap A:E\in \mathcal{E}\}. $$ Show that the $\sigma$-algebra ...
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37 views

Looking for differentiable function $f:\mathbb R \to \mathbb R$ whose derivative is nowhere continuous [duplicate]

Does there exist a differentiable function $f: \mathbb R \to \mathbb R$ such that its derivative $f'$ is nowhere continuous ?
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1answer
35 views

A closed set $F \subset \mathbb{R}$ such that $F, F', F'', F''',\dots $ are all distinct

Let $F \subset \mathbb{R}$ be a closed set. Let $F'$ be the set of the limit points of $F$. Question: Does there exists a set $F$ such that $F, F', F'', F''', ..... $ are all distinct and nonempty? ...
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17 views

A simple question related to One-to-One function and linear operator

I was stuck in one line derivation about the linear operator-related question: Suppose $T$ is linear operator maps from $\mathbb{R}^n$ to $\mathbb{R}^n$. and let $c>0$ be constant. If for all ...
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19 views

Weak subsolution and composition with convex smooth function

I'm trying to solve a problem in Partial Differential Equations by Lawrence C. Evans. It goes like this Assume $u\in H^1(U)$ is a bounded weak solution of \begin{equation}-\sum_{i,j=1}^n ...
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1answer
27 views

For which $s$ is the function $(||x||^{s-2}x_i)^2$ integrable on the unit ball of $\mathbb R^n$?

Initial task is to find out, for which $s$ stands $u=||x||^s \in H^1(\Omega)$, where $\Omega = B(1,0)\subset\mathbb{R}^n$ and $H^1(\Omega)$ is a Sobolev space $W^{1,2}(\Omega)$. As to prove this, we ...
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25 views

Variant of Picard-Lindelof theorem

Question Let $I=[0,a]$ and define the norm $||f||_{\lambda}=\sup_I |e^{-\lambda x}f(x)|$ for $f\in C(I)$. Let $\phi:\;\mathbb{R}^2\to\mathbb{R}$ satify $|\phi(x,u)-\phi(y,v)|\leq\rho |u-v|$ for all ...
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22 views

Sign of a function on interval [0,1]

Let $f(x)=(nx)^{-1+\frac{1}{s}}$ , $g(x)=(nx)^{-\frac{1}{2}}$ with $ 0\leq x \leq 1$ and $n\geq1$ Study the sign of $f-g$
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2answers
54 views

Countable subset and monotonic function

let E be subset of R which has no isloated points(or C does not have any isolated point of E) and C be countable subset of R does there exist a monotonic function on E which is continuous only at ...
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39 views

$\sum \cos(2n)$ Is upper bounded

Proof that $\sum \cos(2n)$ where $n = 1, 2, 3, ...$ is upper bounded. Is ther an easier way than using Moivre's formula? I did try that and failed with complex part.
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29 views

The set composed of domain and codomain of integrable function measure zero

There is this problem which I have constructed a plan to prove, and I am stuck. If anyone could see my plan and tell what is wrong about it I would be very thankful. Let $f: Q \to [0,1]$ be ...
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21 views

Convergence of monotone $f_n:[0, \infty) \rightarrow [0,1]$ to continuous, monotone $g$ is uniform

Let $g: [0, \infty) \rightarrow [0,1]$ be a continuous, monotone increasing function where $g(0)=0$ and $g(x)\rightarrow 1$ as $x \rightarrow \infty$. Also let $f_n:[0, \infty) \rightarrow [0,1]$ be a ...
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1answer
23 views

For $s_n$ a sequence in $\Bbb R$, if $\lim s_n$ defined as a real number, then $\liminf s_n = \lim s_n = \limsup s_n$.

I'm studying some real analysis, and I'm trying to figure out how to prove the following theorem. For the most part, I've got things figured out. I'll post the theorem and my proof (which closely ...
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1answer
29 views

Is it true that a quasiconvex, increasing and continous function, is convex?

Let $f:\mathbb R^n \to \mathbb R$ be a continuous and increasing function. Let $f$ be quasiconvex. Let $f(0)=0$. Can we say that $f(x)$ is convex ? If yes, how do we prove it ? Thank you very much ...
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66 views

Real analysis book suggestion

I am searching for a real analysis book (instead of Rudin's) which satisfies the following requirements: clear, motivated (but not chatty), clean exposition in definition-theorem-proof style; ...
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1answer
13 views

Pre-measure restricted to a set in $\mathcal{A}$.

Definition from Folland-Real Analysis, A function $\mu_0: \mathcal{A} \to [0, \infty]$ will be called a $\textbf{premeasure}$, if (a). $\mu_0(\emptyset)=0$ (b). if $\{A_j\}_1^{\infty}$ is a ...
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18 views

Derivative of Smooth convex function composed with sobolev is sobolev?

This is homework so no answers please Here is what I mean specifically The full problem is $f:\mathbb{R}\to \mathbb{R}$ smooth and convex, bounded $u\in H^{1}(U)$ and $v\in H_{0}^{1}(U)$, is it ...
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47 views

proof $ \sum_{x\geq 1}\frac{-\cos(2x)}{x} $ is convergence?

How to proof that $ \sum_{x\geq 1}\frac{-\cos(2x)}{x} $ is convergence? I guess I am suppose to use Dirichlet test but I am really struggling with it.
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36 views

An inequality about Hermitian matrices

Say one knows the following statement, That for any Hermitian matrix $H$ with eigenvalues $\lambda_1 \geq \lambda_2 ..\geq \lambda_n$ one has, that in any basis, for any positive integers $1 \leq i_1 ...
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35 views

Lebesgue integration in one variable

I have studying the conditions for the existence of the Lebesgue integral. Generally, to show that existence of the integral of a function on an unbounded interval, one can integrate and take ...
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26 views

Orthonormal system

Let $\varphi\in L^2(\mathbb{R})$, prove that $\{e^{2\pi i m x}\varphi(x)\}$ is an orthonormal system iff $$\sum_{n\in\mathbb{Z}}|\varphi(x-n)|^2=1 \ \ a.e \ x$$ How do you prove this. The hint is ...
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1answer
25 views

How can we go from uniform integrable to uniform L1

To be specific, if we have a set of $\{f_n\}$ measurable functions, uniform integrable: For any $\epsilon>0$, we can find a $\delta$, such that for any set E with $\mu\{E\}<\delta$, $|\int ...
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1answer
17 views

Prove that if f is an element of Lip a and a>1 then f is constant

We have that a function $f: [a,b]\mapsto\Bbb R$ satisfies a Lipschitz condition of order $a > 0$ if there is some positive constant $M$ so that $$|f(x_1)-f(x_2)| \leq M|x_1-x_2|^a \qquad \forall ...
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1answer
27 views

Inequality involving Holders Inequalities

Suppose $f\in L^p(\mathbb{R})\cap L^\infty(\mathbb{R})$ for some $p>2$, show that $||f||_{p}\leq ||f||_2^{2/p}||f||_{\infty}^{1-2/p}$ I tried to write $|f|^p=|f|^{\frac{p}{2}}|f|^{\frac{p}{2}}$ ...
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1answer
28 views

Prove that if f is an element of Lip a, then f is uniformly continuous

I am required to prove that if f is an element of Lip a, then f is uniformly continuous. Where Lip means a Lipschitz function. I also have to prove that if f is an element of Lip a and a>1 then f is ...
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1answer
18 views

The set of differentiability of an extension from half-plane to the plane

This question is related to: Differentiability: Partially Defined Functions Consider a real-valued function $f:\mathbb{H}^2\to\mathbb{R}$. Are there some that admit no extension differentiable in ...
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1answer
59 views

Does the statement “$f = 0$ almost everywhere” depend on the measure that is defined?

I know the convention is to use the Lebesgue measure but is there ever a situation where we would interpret "$f(x) = 0$ almost everywhere" by using a different measure? For example, let $f(x) = 1$. ...
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32 views

Convergence of $|f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)}$ on $x \in [0,1]$

I have the following relationship between $f_n$ and $f$ on $x \in [0,1]$ \begin{align*} |f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)} \end{align*} for all $x \in ...
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25 views

Periodic functions and limit at infinity

Why do periodic functions (like $\cos$ or $\sin$ or $\tan$) have no limit at infinity? I can guess that it is because their values don't converge but repeat over and over, but I would like to know ...
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Example of metric space completion

I'm looking for examples of noncomplete metric spaces and their completions. I know of some basic examples like completion of open intervals and rational numbers(both with the reals and p-adic ...
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1answer
25 views

Prove that the antiderivative of an integrable function is both bounded and integrable

Let $f: [a,b] \to \mathbb{R}$ be a bounded function which is also integrable. Define $F: [a,b] \to \mathbb{R}$ by $$F(x)=\int_{a}^xf(t)\ dt$$ To prove that $F(x)$ is also bounded and integrable I ...
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26 views

Complete subspace of continuous function from compact subset [on hold]

Assume $K\in \mathbb{R}$ compact. How to prove that $C^0(K,\mathbb{R})$ is complete. Where $C^0(\mathbb{R},\mathbb{R})$ is the space of continuous f from $\mathbb{R}$.
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25 views

Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.

I have this question : Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so $f$ bounded. I want to know if my proof is valid : If continuous in $x_0$ then : $$\lim_{x \to ...
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80 views

Show that $g(x)=\sqrt{x}$ is uniformly continuous on $[0, \infty)$

I must show that $g(x)=\sqrt{x}$ is uniformly continuous on $[0, \infty)$. I know we have to show that $\sqrt{a-b}\ge \sqrt{a} - \sqrt{b}$ and $\sqrt{a + b} \le \sqrt{a} + \sqrt{b}$ but I don't know ...
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16 views

maximum property of sequence of functions

Let $T>0$, $w:\mathbb R^n\times (0,T]\to \mathbb R$ be continuous and $w$ has a strict lokal maximum in $(x_0,T)$ for some $x_0\in\mathbb R$. Define $$\tilde w(x,t):= w(x,t) - ...
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3answers
27 views

Continuous functions and converging sequences

If $f$ is continuous on $[a,b]$ and $\{ x_n\}$ is a sequence in $(a,b)$, then $\{f(x_n)\}$ has a convergent subsequence. Is it true or false? I think this is false but I am not sure of an ...
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1answer
26 views

Product of a Convergent Series and Bounded Sequence

Let $a_n$ be a bounded sequence and $\sum_{n=1}^\infty b_n$ be a convergent series. Then $\sum_{n=1}^\infty b_na_n$ is convergent. I have found a counterexample to prove it false; If we let ...
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1answer
26 views

How can I show that if a set is bounded, then it's contained in a k-cell?

The set is a bounded subset of R (under the Euclidean metric), and a k-cell is a set of points {x_1...x_k} such that a_j < x_j < b_j for j=1...k. Any ideas on how to show this?
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33 views

Inequality about squareroots [duplicate]

If $a,b\geq 0$ show that $\left| \sqrt{a}-\sqrt{b}\right|\leq\sqrt{\left|a-b\right|}$. WLOG we can assume that $a\geq b$. If one of them is $0$ this is trivial. So assume none of them is $0$. Now, ...
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1answer
16 views

Prove that if $\sum g_k$ converges uniformly on a set $S$ and if $h$ is a bounded function on $S$, then $\sum hg_k$ converges uniformly on $S$

Prove that if $\sum g_k$ converges uniformly on a set $S$ and if $h$ is a bounded function on $S$, then $\sum hg_k$ converges uniformly on $S$ A little confused about this question, would love to ...
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1answer
22 views

Second mean value theorem proof

I am asked to prove the second mean value theorem: Let $f$ and $g$ be defined on $[a,b]$ with $g$ continuous, $f\ge 0$, and $f$ integrable. Then there is a point $x_0 \in (a,b)$ such that $$ ...
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1answer
16 views

ML-inequality for real integrals

For a homework assignment from my analysis class, I was asked to show the following: Let $f\colon [a,b] \to \mathbb{R} $ be Riemann integrable and $|f(x)| \le M$. Define $F(x) = \int_{a}^{x}f(t)dt$. ...