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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.

0
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0answers
47 views

Example of a continuous function

Give an example of two open sets A, B and a continuous function $f:A\cup B\rightarrow\mathbb{R}$ such that $f|A$ and $f|B$ are uniformly continuous but $f$ is not. I have been stuck in this one for a ...
0
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0answers
28 views

Convergence Rate of Smoothing Splines

Consider the regression model with data $(Y_i, x_i)_{i=1}^{n}$ (i.e. real-valued target and univariate predictor x_i) $$ Y_i = f(x_i) + \epsilon $$ with $\epsilon$ iid, $E(\epsilon) = 0$ and $Var(\...
0
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0answers
62 views

Comparing $\sigma$ algebras with topologies

I am learning measure theory from Papa Rudin. I am just trying to capture the ideas conceptually. First of all, how far is measurable sets from topological sets. I guess this question could be ...
2
votes
3answers
35 views

Existence of the derivative at a point is implied by a version of the symmetric derivative plus continuity

This is problem 1.1.1.(ii) on p.10 from Flett's book "Differential Analysis". Variants of the problem have appeared in this forum under the subject of symmetric derivative (e.g., here and here). Flett ...
2
votes
1answer
34 views

Is the set of continuous functions measurable?

Let $X$ be a separable metric space. Then, is $C([0,\infty),X)\in \otimes_{t\in [0,\infty)} \mathscr{B}_X$? I want to write what I have tried at least, but I have no idea how to approach this kind ...
1
vote
2answers
40 views

Statements equivalent to $\sum\limits_{n=1}^{\infty} b_n$ diverging.

Let $b_n$ be a real sequence. I'm trying to figure out if $\sum\limits_{n=1}^{\infty} b_n$ diverging is equivalent to the statement $\forall M\in\mathbb R, \forall\varepsilon>0,\ \exists n_0\in\...
2
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0answers
41 views

Why does evaluation of a two-variable limit fail when using polar coordinates?

The definition of the limit of a two-variable function: $\lim\limits_{(x,y)\to (a,b)}f(x,y)=L\,$ if and only if for all $\epsilon>0$ there exists a $\delta >0$ such that $$0<\sqrt{(x-a)^2+...
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0answers
15 views

How can we apply the non-stationary phase theorem with dependence on large parameter?

I would like to estimate the following integral for long times $T>0$ independent of $\epsilon$ in the limit that $\epsilon \rightarrow 0$ $$G_\epsilon(t) := \frac{1}{\epsilon}\int_0^t f(s,t) e^{\...
2
votes
2answers
32 views

Polynomials with a fixed maximum degree cannot be used to approximate functions uniformly to any desired accuracy?

Let $C[a,b]$ denote the set of all continuous, real-valued maps on the interval $[a,b]$. Let $P_n$ denote the set of all real polynomials on $[a,b]$ which have a maximum degree of $n$. Let $P=\cup_{n\...
-1
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1answer
35 views

If $\{x_n\}_{n=1}^{\infty}$ and $L_S = \lim\limits_{n\to \infty}\sup\limits_{k\geq n} x_k$, show $L_S = \inf\limits_{n\in\mathbb N} a_n$ [on hold]

Please be as critical as possible. Thank you. Since $a_n = \sup \{x_k:k \geq n\}$, $a_n \geq \max\{x_k:k \geq n\}$. Also, $\forall x_k$, $x_k \leq L_S$. Therefore, $L_S = \inf\{a_n:n \in \mathbb N\}...
-3
votes
0answers
17 views

The top 100 people in a list are standing in a circle in clockwise manner. A person starts c [on hold]

The top 100 people in a list are standing in a circle in clockwise manner. A person starts counting from position 1 and disqualifies every second person. For example, the 2nd person is disqualified, ...
5
votes
2answers
250 views

Proving the Borel-Cantelli Lemma

Let $\{{E_k}\}^{\infty}_{k=1}$ be a countable family of measurable subsets of $\mathbb{R}^d$ and that \begin{equation} % Equation (1) \sum^{\infty}_{k=1}m(E_k)<\infty \end{equation} Let \...
0
votes
0answers
23 views

Dimension of smooth manifold

When I prove that $S^1$ is a smooth manifold, I'm considering $S^1$ as a topological subspace of $\mathbb{R}^2$ right? I understood the demonstration that $S^1$ is a one-dimensional smooth manifold, ...
3
votes
1answer
38 views

Supremum of $M = \{ \left \lfloor{\alpha n} \right \rfloor\frac{1}{n}: n \in \mathbb{N}_{>0}\}$

Lets assume the set $M = \{ \left \lfloor{\alpha n} \right \rfloor\frac{1}{n}: n \in \mathbb{N}_{>0}\} $ with $\alpha \in \mathbb{R}, \alpha > 0$. How can I systematically find the supremum of ...
0
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0answers
29 views

Subset of Lebesgue measurable subset of Vitali set is NOT Lebesgue measurable

$B_x = \{y \in [0,1]: x-y \in \Bbb{Q}\}$, $ \varepsilon=\{C \subset [0,1]: \exists x \quad C=B_x\} $. By the axiom of choice we can choose exactly one element of each equivalence class $\varepsilon$ ...
1
vote
1answer
28 views

Hoelder continuity of $\frac yx$ for $x\in (0,1)$ and $0<y<x^2$

I would like to see a proof that the function $$ f(x,y) = \frac yx $$ is Hoelder continuous with exponent $\frac 12$ on the region $$ D:= \{ (x,y): \ x\in (0,1), \ 0<y<x^2 \}. $$ That is, I am ...
10
votes
1answer
107 views

Let $S=\{p(x) \in \mathbb Z[X] :|p(x)| \leq 2^x, \forall x\in \mathbb N\}$. Prove $|S| < \infty$

Question: Let $S=\{p(x) \in \mathbb Z[X] :|p(x)| \leq 2^x, \forall x\in \mathbb N\}$. Prove $|S| < \infty$. Notice this is not true in $\mathbb R[X]$, as $|x-a|\leq2^x $, $a\in[0,1]$ shows. ...
1
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1answer
40 views

Example of function such that integral of $|x(t)|$ is $<+\infty$, and integral of $|x(t)|^2$ is $\infty$

I'm trying to look for an example of a function such that: $$ 1. \displaystyle{\int \limits_{- \infty }^{+ \infty }} \lvert x(t) \rvert \, dt < + \infty $$ $$ 2. \displaystyle{\int \limits_{- \...
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votes
0answers
28 views

Then $X$ is compact , path connected and connected set ?? [on hold]

$X$ be a subset of $\mathbb{R}^{2}$ $X=\{(x,y)|x=0,|y|\leq 1\} \cap \{(x,y)|0<x\leq 1 ,y=\sin(\frac{1}{x})\} $ Then $X$ is compact and path connected?? How to prove its compact I think x=0, y co-...
0
votes
1answer
29 views

If $f$ is continuous on $\mathbb R$ and limits to $\infty$ ,$-\infty$ exist, prove that $f$ is uniformly continuous

Let $f:\mathbb R\to \mathbb R$ be a continuous function such that $\lim_{x\to \infty} \> f(x)$ and $\lim_{x\to -\infty} f(x)$ exist and are finite. Prove that $f:\mathbb R\to \mathbb R$ is ...
0
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0answers
24 views

Prove that transfinite hierarchy of Borel sets will eventually stabilize at some $\lambda < \omega_1$

We define the transfinite hierarchy of Borel sets $\langle {\bf \Sigma}^0_\alpha, {\bf \Pi}^0_\alpha \rangle_{\alpha \in \rm{Ord}}$ as follows: $$\begin{aligned} &\begin{cases} {\bf \Sigma}^...
0
votes
0answers
25 views

Proof of Lusin's Theorem with the characteristic function

Let $m$ be the Lebesgue measure and the set $E$ be Lebesgue-measurable, and $m(E)<\infty$. Prove that for any $\epsilon>0$ there is a compactly supported continuous function $g:\mathbb{R} \to \...
2
votes
2answers
50 views

Lipschitz function differentiable

Let $f:R\to R$ be a Lipschitz function. Suppose $\lim_{n\to \infty}n[f(x+\frac{1}{n})-f(x)]=\lim_{n\to \infty}n[f(x-\frac{1}{n})-f(x)]=0$. Show that $f$ is differentiable on $R$. I think this ...
0
votes
1answer
26 views

$h(x,y)=f(x)f(y)$ is integrable then $f$ is integrable

Let $f$ be a real valued measurable function on $[0,1]$ and let $h(x,y)= f(x)f(y)$ be integrable on $[0,1] \times [0,1]$ . Prove that $f$ is integrable on $[0,1]$. Using Fubini the $x$-section $h_x(y)...
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0answers
21 views

Necessary and sufficient conditions for one sided derivatives to exist

Let $f:\mathbb{R} \to \mathbb{R}$ be a function and $x_0 \in \mathbb{R}$. We denote the left and right hand derivative respectively by $Lf'$ and $Rf'$. If the two exist and match at a point, we say ...
0
votes
1answer
21 views

Derivative bounded above, limit at boundary

Let $f:(0,1)\to \mathbb{R}$ be differentiable with $f'(x)<1,\forall x\ \in (0,1)$. Show: $\lim_{n\to \infty}f(\frac{1}{n})$ exists (could be infinity). First I noticed that $f'$ is not ...
2
votes
2answers
57 views

Prove $\lim_{(x, y) \to (0, 0)} f(x)$ exists if and only if $m + n > 2$

Problem: Let $m, n \in \mathbb{N}$. Show that $\lim_{(x, y) \to (0, 0)} \frac{x^{m}y^{m}}{x^{2} + y^{2}}$ exists if and only if $m + n > 2$. My try: I'm really not too sure about how to prove ...
3
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2answers
32 views

Real Analysis - Absolute Values

Can somebody explain me this step? How did we get rid of absolute values? $||\frac{a_{n+1}}{a_n}|-r| < \epsilon$ implies $|\frac{a_{n+1}}{a_n}|-r < \epsilon$
1
vote
1answer
49 views

Find all the limit points of $\{n/\sqrt{2}+\sqrt{2}/n: n\in \mathbb N\}$

My try. Negative number cannot be the limit point as if take any negative number as a limit point then its nhd will not contain infinite element of set.but what about zero and positive number??? ...
0
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0answers
18 views

Proving associative property of an infinite convergent series

The wording of this question had confused me so much that I had to go to the solutions manual before even attempting the proof. After staring at it for an hour, utterly defeated, I gave in. Prove ...
0
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0answers
32 views

Analysis 1B: Proof that if $f\colon [0,1] \to \mathbf R$ is not continuous, then it is not regulated

Question is basically just what is in the title. I am trying to prove that the function $f\colon [0,1] \to \mathbf R$ being not continuous implies that the function is not regulated over the interval $...
4
votes
4answers
60 views

Disprove or prove using delta-epsilon definition of limit that $\lim_{(x,y) \to (0,0)}{\frac{x^3-y^3}{x^2-y^2}} = 0$ [duplicate]

I want to prove if the following limit exists, using epsilon-delta definition, or prove it doesn't exist:$$\lim_{(x,y) \to (0,0)}{\frac{x^3-y^3}{x^2-y^2}} = 0$$ My attempt: First I proved some ...
1
vote
1answer
18 views

Function Derivable in a interval

Let $f:I \longrightarrow \mathbb{R} $ a function derivable in $I$ where $I$ is a arbitrary interval. If $f'(x)=0, \forall x \in I$ then $f$ is constant. The known statement, or at least for me it is,...
3
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3answers
84 views

Does there exist a function $f$ such that… [on hold]

Is there a function $f:[0, 1]→[0,∞)$ satisfying: $\int^1_0f(x) \ dx=1$,$\int^1_0x f(x) \ dx=\frac{1}{π}$, and $\int^1_0x^2f(x) \ dx=\frac{1}{π^2}$ ? Can I multiply $f$ by some function (maybe $x$ ...
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0answers
31 views

Density of Dynamical System in Function Space

Let $\mathcal{X}\subset C^0(\mathbb{R}^d;\mathbb{R}^d)$ be a non-empty, proper subset which is closed under composition. Under what conditions is the set $\mathcal{X}^{\infty}$ defined by $$ \mathcal{...
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votes
3answers
77 views

Show $\lim\limits_{x\to\infty}f(x) =0$ if $f$ is uniformly continuous and $\lim\limits_{b\rightarrow\infty} \int\limits^b_0|f(x)|dx$ exists [on hold]

Suppose $f:[0,∞) \rightarrow \mathbb{R}$ is uniformly continuous and $\lim\limits_{b\rightarrow ∞} \int\limits^b_0|f(x)|dx$ exists as a real number. Show $\lim\limits_{x→∞}f(x) =0$. I tried a proof ...
0
votes
0answers
45 views

When does a homeomorphism $h:[0,1]\to[0,1]$ preserve Borel sets with Lebesgue measure zero?

Let $h:[0,1]\to[0,1]$ be an order preserving homeomorphism and let $m$ denote Lebesgue measure. I wish to show that given a Borel set $A\subset[0,1]$, $m(A)=0$ if and only if $m(h(A))=0$. Now I know ...
0
votes
0answers
15 views

Lebesgue measure restricted to subsets of the real numbers.

We have the Lebesgue measure $$\overline{\lambda}: \mathcal{B}(\mathbb{R})_\lambda\to [0, +\infty]$$ where $\mathcal{B}(\mathbb{R})_\lambda$ is the completed $\sigma$-algebra of the Borel sets on ...
3
votes
1answer
46 views

Need help checking and verifying that my proofs are correct.

Using these field axioms: 1. [2.][2.] Prove that for all $a,\, b\in\mathbb R$, $ab>0$ if and only if $a>0$ and $b>0$, or $a<0$ and $b<0$. In other words, $ab$ is positive if and only ...
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2answers
22 views

countability of the set of discontinuities

Is the set of discontinuities (points where our function is discontinuous) for a monotone function countable even if it's defined on an unbounded interval if so why?
0
votes
1answer
20 views

Continuity of functions examples

I need to give explicit examples of two functions $f$: [0,1]$\in\to\mathbb{R}$ and $g$: (1,2]$\in\to\mathbb{R}$ such that $f$ and $g$ are continuous, but the function $h$: [0,2]$\in\mathbb{R}$ ...
3
votes
0answers
39 views

Is the set of points where two measurables functions are equal measurable?

Let $(\Omega,\mathscr{F},P)$ be a measure space and $T$ be a metric space and $f,g:\Omega\rightarrow T$ be measurable functions. Then, is the set $\{w\in \Omega:f(w)\neq g(w)\}$ measurable? I could ...
0
votes
1answer
66 views

Peano theorem — application to Cauchy problem

How do we prove existence of this Cauchy problem $$ \begin{cases} y'= f(x,y)= y \ln|y|\\ y(x_0)=y_0 \end{cases} $$ using Peano theorem? I try this: First, we have that $$ \dfrac{\partial f}{\...
1
vote
1answer
43 views

Show that $\sqrt{ab}$ is the limitpoint of the nested interval

I already know that positve $a,b$ we have $\frac{a+b}{2}\geq\sqrt{ab}\geq\frac{2ab}{a+b}$ Given, $0<a<b$ and $I_n=[a_n,b_n]$ with $I_1=[a,b]$ and $I_{n+1}=[\frac{2a_nb_n}{a_n+b_n},\frac{a_n+...
1
vote
2answers
56 views

Let $f:[0,\infty)\to\mathbb{R}$ be a continuous function with given conditions. Prove $\lim_{x\to\infty}f(x)=0$

Let $f:[0,\infty)\to\mathbb{R}$ be a continuous function s.t. $\forall ~x \in[0,\infty), f(x) \neq0 $ and $ \forall ~ \varepsilon>0 ~ \exists ~ x\in[0,\infty)$ $s.t. 0< f(x) <\varepsilon$. ...
0
votes
3answers
46 views

If $0<a<b$ then $\frac{b-a}{2(a+b)}<\frac{1}{2}$?

I don't know how I can prove this or whether it is right or not, but I want to make an estimate of the term $$\frac{(a_n-b_n)^2}{2(a_n+b_n)}.$$ I have so far only figured out that it is equal to $...
2
votes
3answers
289 views

Question about the Fundamental Theorem of Calculus with variable integration limit

If $F(x)=\int_2^{x^3}\sqrt{t^2+t^4}dt$ a.) The integral of $F(x)$ is $3x^2\sqrt{x^2+x^4}$. b.) The derivative of $F(x)$ does not exist. c.) $F'(x)=3x^2\sqrt{x^6+x^{12}}$. I can't seem to find the ...
0
votes
0answers
24 views

Prove the continuity of a function in a normed space

Let $(X,||·||)$ be a normed space over a field $\mathbb{K}$, where $$||x||=\max\{||x_1||,||x_2||\}$$ for any $x=(x_1,x_2)\in\mathbb{K}\times X$. Show that the mapping \begin{equation*} \begin{...
0
votes
0answers
15 views

Less than linear growth speed - nested intervals that converge to an interval and not a point

To call something a nested intervall the length of the intervals have to go to zero and each successiv inervall must be within the previous. so if $[a_n,b_n]$ is a nested interval we would need a ...
-2
votes
3answers
78 views

some confusion about singleton set?

Is the set $\{0\}$ is closed in $(\mathbb{R} , |.|)$ ? where $|.|$ denotes the usual metric on $\mathbb{R}$ My attempt : yes , because i think $\{0\}$ is not open for all $x> 0, (x- \...