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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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0answers
11 views

Showing differentiability of $g(x)=\begin{cases}\frac{f(x)}{x},&\text{$x\neq0$}\\f'(0),&x=0\end{cases}$ given that $f(0)=0$

Let $f$ be a twice-differentiable function of $\mathbb R$ with $f(0)=0$. Define $$g(x)=\begin{cases}\frac{f(x)}{x},&\text{$x\neq0$}\\f'(0),&x=0\end{cases}$$ Prove that $g$ is a differentiable ...
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0answers
10 views

Unique Partition of $\{1, \dotsc, n\}$ by Ultrametric Property

I'm currently working on a proof which states (not the main claim, just a property used later on in the proof): Let $d=(d_{ij})$ be an ultrametric on the set $\{1, \dotsc, n\}$ (i.e. that $d$ is a ...
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4answers
32 views

Upper Bound of $x(1-x)$ for $x\in [0,1]$?

Is there a good upper bound for $x(1-x)$ if $x\in [0,1]$?
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1answer
28 views

Do $f_n \to f$ almost surely and $0<\int f<\infty$ imply $\int f_n \to \int f$?

Let a sequence of Lebesgue integrable functions $f_n$ converge to $f$ almost surely and $0 < \int f < \infty$. Does this imply $\int f_n \to f$? If the condition $0 < \int f$ is removed there ...
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1answer
19 views

Compactness of intersection of a compact set and an open set

If $K \subset E_1 \cup E_2$, where $K$ is compact and $E_1, E_2$ are disjoint open subsets of a topological space, is $K \cap E_1$ compact? Is that always the case if $E_1, E_2$ are not disjoint? I'...
1
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1answer
22 views

Does $f_n \to f$ pointwisely imply $\int f_n \to \int f$ for conditionally Riemann integrable functions?

Let $f_n:\mathbb{R}^n\to\mathbb{R}$ be a sequence of conditionally (improper) Riemann integrable functions that pointwisely converges to $f:\mathbb{R}^n\to\mathbb{R}$ which is also conditionally ...
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2answers
33 views

Lebesgue integral question

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a measurable function. Show that if $f$ is continuous and fix $x_0\in \mathbb{R}$, then $\lim_{n\rightarrow\infty}n\int_{x_0}^{x_0+1/n}fdm=f(x_0)$. Hint: ...
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0answers
8 views

How to prove this result somewhat similar to Du Bois-Reymond's Lemma?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded connected set with smooth boundary. Suppose also that for each index $i, j = 1, \dots, n$, $f_{ij}$ is a smooth function. If for every $v\in C^...
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0answers
19 views

Using Picard-Lindelof to find a solution to $y'(t,y(t))=t+\sin(y(t))$ where $y(2)=1.$

Consider the initial value problem $y'(t,y(t))=t+sin(y(t))$ with initial condition $y(2)=1$. Find the largest interval $\mathcal{I}\subset \mathbb{R}$ containing $t_0=2$ such that the problem has a ...
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1answer
27 views

If a sequence converges, do its tail terms equal its limit?

A sequence $(x_n)$ in a metric space $X$ is said to converge if there is a point $x \in X$ such that for every $\epsilon > 0$ there is an integer $N$ such that $n \geq N$ implies that $d(x_n, x) &...
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1answer
18 views

Uniformly approximating $f\in \mathcal{C}([1,\infty))$ with polynomials where $\underset{x\rightarrow +\infty}{\lim} f(x)=a$

Suppose $f\in \mathcal{C}([1,\infty))$ and $\underset{x\rightarrow +\infty}{\lim} f(x)=a$. Show that $f$ can be uniformly approximated on $[1,\infty)$ by functions of the form $g(x)=p(1/x)$, where $p$ ...
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0answers
22 views

Sum of a darboux function and a continuous function

Could someone find an example where the sum of a darboux function and a continuous function is no longer darboux(have intermediate value property)? I am given to understand this is possible in my ...
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0answers
26 views

Multiplying conditionally convergent series. Is this valid?

Normally when I see this form of the product of infinite series: $$ \left(\sum_{n=0}^\infty a_n\right)\left(\sum_{m=0}^\infty b_m\right) = \sum_{n=0}^\infty\sum_{m=0}^\infty a_nb_m $$ I see the ...
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1answer
20 views

Prove that the liminf of a sequence is equal to its smallest subsequential limit

$\ a:= \liminf_{n\to \infty}$ $a_n$ = $lim_{n\to \infty}$ $inf_{k\geq n}$ $a_k$ $a$ $\in$ $R$ I am trying to show this by using cases and assuming the equality doesn't hold- to come to a ...
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1answer
11 views

separated and dense in locally compact is discrete?

It is true that if $A\subset \mathbb{R}$ $\delta$-separated and $\epsilon$-dense in $\mathbb{R}$ then $A$ is discrete? (This because of $\mathbb{R}$ is locally compact? I ask this since $\mathbb{Z}$ ...
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0answers
19 views

Step-by-step convergence of infinite product

total amateur here. Is it true that if an infinite product is equal to some real number $x$ the infinite product converges to $x$? Second question: What is the convergence test for an infinite ...
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1answer
38 views

Real Analysis: Function Continuity

Let $A, B \subset \mathbb{R}$, let $f : A \rightarrow B$, and let $x_0 \in A$. Then $f$ is continuous at $x_0$ if and only if $\lim_{x \rightarrow x_0} f(x) = f(x_0)$. I proved the first part $f$ ...
5
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1answer
45 views

Value of operator norm when $\mathcal{T}f(x)=\int^{x}_{0} f(t)dt$

Let $\mathcal{C}$ be the space of continuous functions on $[0,1]$ equipped with the norm $\|f\|=\int^{1}_{0}|f(t)|dt$. Define a linear map $\mathcal{T}:\mathcal{C}\rightarrow \mathcal{C}$ by $$ \...
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0answers
6 views

Iteration of a sequence

I am reading a paper on Piecewise Convex Transformations. I got an inequality as : $||P_{\tau} f||_{\infty} \leq {\frac{1}{\alpha}|| f||_{\infty} + C||f||_{1}}$ After n iterations, we obtain : $||P_{...
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1answer
9 views

Uniformly Bounded and Weakly compactness.

Is a uniformly bounded sequence weakly compact in $l_1$? If Yes, What is the reason?
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2answers
23 views

When does subsequence convergence imply convergence?

I know that for a sequence $\{x_n\}$, if $\{x_{2n}\}$ and $\{x_{2n+1}\}$ converge to the same limit, then $\{x_n\}$ converges and to the same limit. My question is if I know that from any $x_i$ I can ...
0
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0answers
7 views

Find the sup(S) ,inf(S) where S = { x belong to [-1,4] |sin(x)>0} [on hold]

Let S = { x belong to [-1,4] |sin(x)>0} how to solve the problem I need a help to find sup(S) ,inf(S)
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2answers
16 views

Does we have separation theorem for closed subsets in a topological vector space?

In Rudin's Functional Analysis, page $10,$ he stated the following separation theorem for topological vector space. Theorem $1.10:$ Suppose $K$ and $C$ are subsets of a topological vector space $X,$...
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0answers
25 views

Expanding two iterated limits into quantifiers?

I have always had a little trouble fully understanding iterated limits. So I thought it might be a good exercise to expand it in quantifiers. Suppose we want to expand $\displaystyle \lim_{x \to a} f'...
2
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0answers
24 views

How many odd and convex functions exist in $R^n$

How many odd and convex functions $f: R^n \to R $ exist ? I guessed just 1, namely $f(x)=\langle a, x \rangle $ where $a$ is a fixed $n-$vector. But I couldn't either prove it or provide counter ...
2
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1answer
22 views

How many Convex and Odd functions exist?

During lunch this question popped on mind. How many odd and convex functions $f: R \to R $ exist ? I guessed just 1, namely $f(x)=ax$ where $a$ is a fixed real number. But I couldn't either prove ...
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3answers
53 views

For what $x>0$ series $\sum_{n=1}^{\infty} x^{1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}…+\frac{1}{\sqrt n}}$ is convergent?

For what $x>0$ series $\sum_{n=1}^{\infty} x^{1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}...+\frac{1}{\sqrt n}}$ is convergent? By logarithmic test, $$ \lim_{n\rightarrow \infty}\left(n \log\frac{u_n}{u_{...
2
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1answer
31 views

Prove that if $f^2(x)$ is Lebesgue integrable on E, so is $f(x)$.

Please check my proof, thank you. Let $f(x)$ be a nonnegative and measurable function defined on the set $E$ with $m(E) < \infty$. Prove that if $f^2(x)$ is Lebesgue integrable on E, then ...
1
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4answers
44 views

Show that the following inequality holds when $x>0$

$\require{cancel}$ Show that the following inequality holds for $x>0$ $$1+\frac{x}{2}-\frac{x^2}{8}<\sqrt{x+1}<1+\frac{x}{2}.$$ I proceeded as follows $$\sqrt{x+1}=1+\frac{x}{2}-\frac{...
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1answer
15 views

Codimension of subspace of sequences that converges to $0$

Let $\bar C =$ be the double converges sequences ($(a_n)^\infty_{n=-\infty}$), and $\bar C_0$ be the double converges sequences where the limit is $0$ for both $-+\infty$. What is $codim(\bar C_0)$? ...
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2answers
39 views

Additive inverse of infinity [on hold]

What example demonstrates that a definition $$ \infty-\infty=\infty+\big( -\infty \big)=0 $$ necessarily invokes a contradiction?
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2answers
30 views

Is any countable subset of an uncountable set closed?

Consider uncountable nonempty $X\subset\mathbb{R}^k$. I wonder if I choose $E\subset X$ such that $E={p_1,p_2,...}$ countable and infinite, then $E$ must be closed in X. Also, if I choose such set in ...
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1answer
26 views

Does the term of a convergent infinite sum create a convergent infinite product with 1 + term?

Let $0<\sum_{n=k}^\infty f(n)<\infty$ where $f(n)>0,n\ge k$. Then is the following guaranteed to be true: $$0<\prod_{n=k}^\infty(f(n)+1)<\infty$$ For example: $$f(n)=\cfrac{1}{n^2}$$ $$...
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0answers
7 views

Convergence of Expectation of norm of sub-gaussian random vector

1.We know that if $X=(X_1,...,X_n)$ be a random vector with independent sub-gaussian coordinates $X_i$ that satisfy $EX_i^2=1$, then $$||||X||_2-\sqrt{n}||_{\psi_2}\leq CK^2$$ where $K=max||X_i||_{\...
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0answers
7 views

Integral convergence $p$-test with complex exponents

It is a well-known result that the integral $$ \int_0^1 \frac{1}{x^p} \, dx $$ converges if and only if $p < 1$. Does the analogous result for complex exponents hold? That is, does the integral $$ ...
2
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1answer
33 views

Factorization for $e^{\lambda x}$

Let $\lambda, x$ be real numbers. Why can't we factorize $$e^{\lambda x}=f(\lambda)g(x)$$ for some functions $f$ and $g$?
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2answers
39 views

Is the set of zeros of $f(x)=x^2\cos(\frac{1}{x})$ is connected over $[0,\frac{2}{\pi}]$?

Let $f : \left[0,\frac{2}{\pi}\right] \rightarrow \mathbb{R}$ be defined by $f(0)=0$ and $f(x)=x^2\cos\left(\frac{1}{x}\right)$ for $x\neq 0$. This function is continuous on $\left[0,\frac{2}{\pi}\...
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2answers
42 views

Show that the unit sphere in $\mathbb{R}^3$ is pathwise(=arcwise) connected?

I know this question has been already asked but no one does not address the definition of pathwise(=arcwise) connectedness as the following: A set $C$ in metric space $(M,d)$ is pathwise(=arcwise) ...
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1answer
57 views

If $\sum_{n=1}^\infty a_n$ converges, then $a_n \rightarrow 0$. [on hold]

I know that this statement is true. However, is the converse of the statement true? If not, what is an example that shows that the converse is not true?
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1answer
13 views

Real analysis- neighbourhood of sets [on hold]

Let I=(a,b) where a,b in R express the interval as a neighbourhood
2
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0answers
32 views

The convergence of an improper integral

Let us find all values of $\alpha$ such that the following improper integral converges: $$I(\alpha):=\int_{1}^{\infty} \frac{ e^{\dot{\imath}(x^2-2x)}}{x^{\alpha}}\,dx$$. When $\alpha>1$, $I(\...
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votes
1answer
23 views

No function continuous at rational points and discontinuous at irrational points in $[0,1]$

Let $C_f$ and $D_f$ mean sets where a function is continuous and discontinuous. I’m trying to prove there is no function $f:[0,1] \to \mathbb{R}$ such that $C_f = [0,1] \cap \mathbb{Q}$ and $D_f = [0,...
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1answer
31 views

Does L'Hopital's Rule extend to $x \rightarrow \infty$ and $L= \infty?$

The following is given: Suppose that $f$ and $g$ are continuous functions on the interval $(a,b)$ and that $$\lim_{x \rightarrow b} \frac{f(x)}{g(x)} = \frac{0}{0}.$$ If $g(x),g'(x) \neq 0$ for all $...
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2answers
54 views

Tricky limit algebra

Could someone please help me figure out the following algbera? I really don't understand any of the steps. $$ \begin{align*} \lim_{x\to 0} \frac{1-\sqrt{1-x^2}}{x} &= \lim_{x\to 0} \frac{(1-\...
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1answer
8 views

Finite outer measure $\mu$ on a particular set implies the existence of another set such that outer measure and measure are equal on respective sets

Let $(X, \mathcal{A}, \mu)$ be a measurable space with a $\sigma-$finite measure $\mu$, while $(X, \mathcal{A^{*}}, \mu^{*})$ is the completion of the underlying space. Show: For $B \in \mathcal{A^...
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0answers
17 views

Property of supremum for any real exponent

For any $\alpha>0$ and $f:\Omega\to(0,\infty)$ bounded, is it true that $\sup f^{\alpha}\leq (\sup f)^\alpha$? Also what about $\alpha<0$. I know for $\alpha\in\mathbb{N}$, it holds. But for ...
1
vote
1answer
24 views

Hint for this equicontinuity/bounded question

Let $X$ be a compact metric space. Prove that an equicontinuous subset of $C(X)$ is uniformly bounded if it is pointwise bounded. I'm not looking for the answer. I know all the relevant definitions ...
0
votes
1answer
31 views

How to determine the period of composite functions?

For exemple, take :$f\left(x\right)=$cos$\left(2x\right)\cdot \:$sin$\left(3x\right)$. Period of cos$\left(2x\right)$ is $\pi$ and that of sin$\left(3x\right)$ is $\frac{2\pi }{3}$. But why is the ...
0
votes
2answers
18 views

$\sigma$-finite measure $\mu$ so that $L^p(\mu) \subsetneq L^q(\mu)$ (proper subset)

I'm looking for a $\sigma$-finite measure $\mu$ and a measure space so that for $1 \le p <q \le \infty$ $$L^p(\mu) \subsetneq L^q(\mu)$$ I tried the following: Let $1 \le p <q \le \infty$ ...
-1
votes
0answers
18 views

Estimate of the distance between a point and a closed real subset. [on hold]

Couldn't really find any idea for solving this... Do you have any? https://i.stack.imgur.com/Pr9Km.png