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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.

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2 views

A property of a singular increasing function on [a, b]

My question is from Royden's Real Analysis (4ed) Chapter 6 Problem 54 Part(i): Let $f$ be a singular increasing function on $[a, b]$. Use the Vitali Covering Lemma to show that $f$ has the ...
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2answers
45 views

Is this function bounded on $[a,b]$?

I need to show that $f:[0,1] \rightarrow \mathbb{R}$ is bounded on $[a,b]$ if for all $c \in [a,b]$, $\lim\limits_{x \rightarrow c} f(x)$ exists. I tried using the definition of limit, but I don't ...
3
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0answers
36 views

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

Let $S=\{p(x) \in \mathbb Z[X] :|p(x)| \leq 2^x, \forall x\in \mathbb N\}$. Find $|S|$. This is a follow-up question to Let $S=\{p(x) \in \mathbb Z[X] :|p(x)| \leq 2^x, \forall x\in \mathbb N\}$. ...
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0answers
15 views

How to prove the finiteness of lower semi-continuous function?

Let $C$ be a convex set, and let $f$ be a finite lower semi-continuous convex function. In addition, we have $f$ is bounded above on every bounded subset of $\text{ri}~C$, where $\text{ri}~C$ is the ...
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0answers
35 views

Real Analysis: Cauchy-Schwarz inequality proof

Appearing in Tao's Analysis II, this proof contains some series manipulations. ($\sum_{i=1}^n a_{i}b_{i}$)$^2$ + $\frac{1}{2}$$\sum_{i=1}^n \sum_{j=1}^n (a_{i}b_{j}-a_{j}b_{i}$)$^2$ = $(\sum_{i=1}^n ...
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0answers
26 views

continuous but non-differentiable function having countable points of non-differentiablity

Consider function $f:\mathbb{R}\to\mathbb{R}$ satisfying $|f(x)-f(y)|\lt 4321|x-y|$ for all real numbers $x$ and $y$. Show that there exists a function $f(x)$ such that $f$ is continuous but is non-...
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0answers
14 views

Partially Non-Computable Numbers

Could there be a number say: A = $0.a_1a_2\cdots a_n$->$a_k \cdots$ where say {$a_1a_2$} and {$a_n$->$a_k$} are computable parts of A. Yet every thing else is not computational. I imagine a scheme ...
2
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1answer
32 views

Prove $h(y) = \int_{\mathbb{R}} f(x,y) dx$ is continuous if $f(x,y)$ is continuous for every fixed $x$ and $f(x,y)$ is integrable for every fixed $y$.

Define a function $f : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$. Suppose that for every fixed $x \in \mathbb{R}, f(x,y)$ is continuous. And suppose that for every fixed $y \in \mathbb{R}, ...
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0answers
15 views

Real number pattern (non-computable numbers)

supposedly the real numbers have a pattern like: [ABABA] where A represents rational numbers and B is irrationals s.t. $A0<B0<A1<B1<A2$. Is there a similar pattern which includes non-...
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1answer
32 views

Identify the limit or show that it does not converge.

Definition 1. The sequence $(a_n)$ in a metric space $(X,d)$ converges to the limit $x\in X$ provided that for each positive real number $\epsilon$, there exists a natural number N so that whenever $n&...
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1answer
15 views

Trying to understand a proof where disjoint decomposition of $\mathbb R^n$ is used.

I am trying to understand what a disjoint decomposition is, specifically in the following proof of Example IV.8 (the last paragraph): I don't understand how they got the last line of Example IV.8 ...
3
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2answers
70 views

Provide example of two series that both diverge but $\sum\min\{a_n,b_n\}$ converges

I've posted the solution for this problem and I'm trying to understand this. In the end of the solution provided it says to continue this process. So, do we hold $a_n$ to be $\frac{1}{n^2}$ and $b_n$...
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3answers
51 views

When $x$ tends to infinity then find limit

Suppose $$f(x)=(\ln(x))^2$$ Then as $x$ tends to $\infty$, find the limit of $$f(x+1)-f(x)$$ (if it exists). $$\begin{align} f(x+1)-f(x)&=(\ln(x+1))^2-(\ln(x))^2 \\ &=(\ln(x(1+1/x)))^2-(\...
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2answers
45 views

Show $\inf(k+A)= k + \inf(A)$

Let $A \subseteq \mathbb { R }$ be a nonempty set which is bounded from below in $\mathbb { R }$ . Given a fixed number $k \in \mathbb { R } ,$ show that the set $k + A : = \{ k + a : a \in A \}$ is ...
2
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2answers
32 views

Proving the infimum of a set?

$A : = \left\{ 6 - \frac { 1 } { n } : n \in \mathbb { N } \right\}$ I know 5 is a lower bound for A I want to use the Archimedean Property to show 5+Epsilon is not a lower bound for all epsilon > 0 ...
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0answers
26 views

Integration of forms over manifolds

Spivak start the section 'Stokes theorem on manifolds' of his book 'Calculus on manifolds' with the following definition: If $\omega$ is a $p$-form on a $k$-dimensional manifold with boundary $M$ ...
4
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1answer
120 views

Algebraic proof that two statements of the fundamental theorem of algebra are equivalent

Students studying the fundamental theorem of algebra in high school are probably familiar with the statement that goes something like the following. Every non-zero, single-variable, degree $n$ ...
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1answer
65 views

Roots of a polynomial in $x$ and $e^x$

I have equations of the following form: $$(p_1(y) + p_2(y) x)^2 + p_3(y) = 0$$ Where $x \in \mathbb{R}$, $y = e^x$ and the $p_i$ are polynomials with real coefficients. Is there any way to say ...
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1answer
27 views

Proof of formula for the arc length

I've read the proof of formula for arc length . I wonder why the function has to have continuous derivative (According to the Stewart Calculus book). I mean in which part of the proof we used this ...
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0answers
28 views

Product rule for derivatives in $R^n$

I'm trying to understand the proof under the product rule of the book, "The elements of real analysis (2nd ed.) by Robert G. bartle" [page 361] Here we are going to use the fact that: If $f:R^p\...
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1answer
36 views

Problem 4 part (a) from Stein's Real Analysis

Let $f$ be a bounded function on a compact interval $J$, and let $I(c,r)$ denote the open interval centered at $c$ of radius $r>0$. Let $osc(f,c,r)=\sup|f(x)-f(y)|$, where the supremum is taken ...
0
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1answer
55 views

Parseval's Theorem Question

Show using Parseval's Theorem that: $$ \sum_{n=-\infty}^{\infty}\frac{1}{(n+a)^2}=\frac{\pi^2}{\sin^2(\pi a)} $$ I've tried to think about ways to solve this but haven't got anywhere. I must be ...
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0answers
44 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 ...
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0answers
24 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(\...
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0answers
30 views

How far is measurable sets from topological sets

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
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3answers
27 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
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1answer
33 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 ...
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2answers
39 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
29 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
8 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^{\...
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2answers
31 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\...
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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\}...
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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, ...
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2answers
245 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 \...
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0answers
22 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
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1answer
37 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 ...
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0answers
28 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 $\varepsilon$ ...
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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 ...
8
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1answer
78 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. ...
0
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1answer
28 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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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-...
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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}^...
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0answers
23 views

Optimizing the ratio of two convex functions, with some special properties

I have a smooth function $f(x) = \frac{g(x)}{h(x)}$ that is the ratio of two smooth convex functions $g$ and $h$. It is known that $f(x)$ has a global minimum, achieved at a unique point $x_0$. Also ...
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
48 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)...
0
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0answers
20 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
54 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 ...