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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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1answer
20 views

Prove that $\mu((0,\infty))=0$.

Suppose $\mu$ is a measure on the Lebesgue measurable subsets of $\mathbb{R}$ and assume that there is a $K\geq 0$ such that for all $n\in\mathbb{N}$, we have $$\displaystyle\int_\mathbb{R} e^{nx}\ d\...
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3answers
16 views

Infimum and supremum for $\{ x \in \Bbb R\mid x^2-2x-1 < 0 \} $

I've been asked to obtain the infimum, supremum, minimum and maximum for: $\{ x \in \Bbb R \mid x^2-2x-1 < 0 \} $ and $\{x^2-2x-1\mid x \in \Bbb R\} $ So for the first, I used the quadratic ...
1
vote
1answer
17 views

Continuous functions, null sets and Lebesgue measurable sets

so im trying to prove that if i have a continuous function then f transforms null sets in null sets if and only if f transform Lebesgue measurable sets in Lebesgue measurable sets. Anyone has got some ...
0
votes
1answer
14 views

Show that $64xy=-1$

If $x=\cos(2a)\cos(5a)\cos(6a)$ and $y=\cos(a)\cos(3a)\cos(9a)$, where $a=\frac{\pi}{13}$, then show that $64xy = -1$. I'm trying to use $$\cos(A)\cos(B)=\frac{\cos(A+B)+\cos(A-B)}{2}$$ then use ...
1
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2answers
25 views

Can dominated convergence justify commuting two limits?

I am evaluating an expression of the form: $$\lim_{a\to 0^+}\,\,\lim_{b\to 0^+}\,\,\sum_{n=1}^\infty\int_{-\infty}^{+\infty}f_n(x;a,b)\,dx.$$ Suppose I can find dominating functions $F_n(x)$ such ...
0
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0answers
16 views

Using Archimedes-Riemann Theorem to prove integrability

Here is an excerpt from a book I am reading. There is one portion that I don't understand, which I explain afterwards: For the rectangle $\mathbf{I} = [0, 1] \times [0, 1]$ in the plane $\mathbb{R}^{...
3
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0answers
23 views

How to show that two trigonometric polynomials of degree $n$ combined have at most $2n$ zeros?

I am already aware of this question: Prove the following trigonometric polynomial has $2n$ zeros But it's not the same. Let be $P(x) = \sum_{k=0}^{n} a_k \cos (kx)$ and $\tilde{P}(x) = \sum_{k=0}^n ...
3
votes
1answer
34 views

If $f:[0,1]\to \mathbb{C} $ be continuous with $f(0)=0$ and $f(1)=2$, then $|f(t_0)|=1$ for some $t_0 \in [0,1]$

Question: Let $T=\{z\in \mathbb{C}:|z|=1\}$ and $f:[0,1] \to \mathbb{C}$ be continuous with $f(0)=0$, $f(1)=2$. Show that there exists at least one $t_0$ in $[0,1]$ such that $f(t_0)$ is in $T$. ...
4
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1answer
42 views

Dense subset of $\mathbb R$

Let us consider an enumeration of $\mathbb Q$ as $\mathbb Q=\{r_1, r_2, \cdots \}$. Now let $y_n= n+r_n $;$\forall n \in \mathbb N$. Let $A=\{y_n| n\in \mathbb N \} $. Is $A$ dense in $\mathbb R$?
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0answers
14 views

Let $\lim_{t\to t_0}\phi(t) = a$. Prove that $f(x) = \mathcal{o}(g(x)) \implies f(\phi(t)) = \mathcal{o}(g(\phi(t)))$

Let: $$ \lim_{t\to t_0}\phi(t) = a $$ where $\phi(t)\ne a$ and $t\ne t_0$ in the neighbourhood of $t_0$. Prove that: $$ \begin{align*} f(x) \stackrel{x\to x_0}{=} \mathcal{o}(g(x)) &\implies ...
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2answers
34 views

Convergence of power series beyond radius of convergence?

In rudin analysis text book, 3.44 theorem , it says about the convergence on the boundary of the circle of the power series, the theorem is roughly:- Suppose the radius of convergence of $\Sigma ...
0
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1answer
32 views

If $T$ is a topological isomorphism, then so is $T^*$

The question comes from the following link on page 25: https://www.ucl.ac.uk/~ucahad0/3103_handout_3.pdf They prove $(T^{-1})^*=(T^*)^{-1}$, but I don't see how it proves $T^*$ is a topological ...
1
vote
1answer
17 views

For $g(x) = \int_0^\infty (x+y)^{-1} f(y) \, dy$ show $|g'(x)| \le c_p \frac{1}{|x|^{1+1/p}} \lVert f \rVert_{L^p}$

Q: For $x > 0$ and $f \in L^P(0,\infty), 1 \le p < \infty$, \begin{align*} g(x) &= \int_0^\infty (x+y)^{-1} f(y) \, dy \\ \end{align*} Show that $g(x)$ is continuous and in fact ...
0
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0answers
78 views

How I can show that $\cos(2π/13)\cos(5π/13)\cos(6π/13)=\frac{\sqrt13-3}{16}$

I need help to prove that $\cos(2π/13)\cos(5π/13)\cos(6π/13)=\frac{\sqrt13-3}{16}$ Without complex method I'm try ... Let x=$\cos(2π/13)\cos(5π/13)\cos(6π/13)=\frac{\sqrt13-3}{16}$ Idon't know how I ...
2
votes
1answer
32 views

Does the Lebesgue measure on the segment $y=x$ represent this distribution?

Set $\Omega=(-1,1)^2$. Consider the following measure on $\Omega$: $\mu(A)=m(A \cap L)$, where $L=\{ (x,x) \, | \, -1 < x < 1\}$ (the segment of the line $y=x$ in $\Omega$) , and $m$ is the ...
0
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0answers
12 views

Proving $\arg$ restricted to an open subset of $S^1$ is smooth

Let $U$ be an open subset of $S^1\subset \mathbb{R}^2$. Define $\theta:U\to \mathbb{R}$ by $\theta(x,y)=\arg(x+iy)$ where $\arg$ is the principal argument. I want to prove that $\theta$ is smooth (i.e....
3
votes
0answers
10 views

Can we perturb a map to have distinct singular values?

Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \text{GL}_n^{-}$ be smooth. ($\text{GL}_n^{-}$ is the set of $n \times n$ real matrices with negative ...
1
vote
0answers
51 views

How to solve the given limit?

$$\lim_{n \rightarrow \infty}n^2 \int_{0}^{1} \frac{1}{(1+x^2)^n}dx$$ How to solve this,what should be our first approach?
0
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1answer
24 views

What is the definition of this $L^p$-space?

I am reading a book by E. Zehnder and I am confused about an $L^p$-space he is using. What is the definition of the space $$ L^p(S^1,\mathbb{R}^{2n}) $$ Thank you for your kind help.
1
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0answers
30 views

Prove that $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous iff $f^{-1}(O)$ is open for each open set $O$

How would you proceed in the below, or have I led myself to a dead-end? Let $O$ be an open set from $\mathbb{R}$. $O \subset \mathbb{R}$ Let $D \subseteq \mathbb{R}$, be the domain of $f$. i.e. $f : ...
0
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0answers
21 views

I have seen the definition of pointwise convergence, i read through logic scripts but still do not know how to read it properly.

it is not only about the definition, i want to know how to read those logicial explanations properly. It says For all epsilons and (?) for all x there exists N for all n>N: ... this does not sound ...
0
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0answers
44 views

Finding inverse function for g

A function $g$ is given by $g(x) = x + f(x)$ , where $f:[0,1]\rightarrow [0,1]$ and $g:[0,1]\rightarrow[0,2]$ and $f(\sum_{n=1}^{\infty}\frac{a_n}{3^n}) = \sum_{n=1}^{\infty}\frac{b_n}{2^n}$ . How to ...
0
votes
2answers
22 views

How can I prove that $f\in L^p\cap L^q\implies f\in L^r$ for all $r\in [p,q]$?

I have a result that says that since $f\in L^p(\mathbb R)\cap L^q(\mathbb R),$ we have that $f\in L^r(\mathbb R)$ for all $r\in [p,q]$. I don't really know how to prove this. I know that if $r\in [p,q]...
0
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2answers
48 views

Do not understand: “A real function $f$ is continuous if $f^{-1}(O)$ is open for each open set $O$”

I am looking at Measure, Integral and Probability book and that is what they say (about the title). Although, I fail to understand this. If we take a function $f(x) = x^2$ and restrict its domain to $...
1
vote
1answer
25 views

Poisson equation $-\Delta u = 1$ with Dirichlet condition

Let $R > 0$. Determine the radial solution of the problem \begin{align} - \Delta u(x) & = 1 \text{ if $|x| < R$}\\ u(x) & = 0 \text{ if $|x| = R$} \end{align} We know the fundamental ...
0
votes
0answers
30 views

For an increasing function $f$, if $f(y_n)-f(x_n) \to 0$ , then $f$ is continuous at $0$

Question: Let $f: \mathbb{R} \to \mathbb{R} $ be an increasing function. Suppose there are sequences $(x_n)$ and $(y_n)$ such that $x_n<0<y_n$ for all $n\geq 1$ and $f(y_n)-f(x_n) \to 0$ as $n \...
0
votes
1answer
47 views

Convergence of $\sum_{n=2}^{\infty}\left(\frac{\log n}{\log(\log n)}\right)^{-\log n/\log(\log n)}$

Initially, I need to prove, that $$\forall \lambda > 0 \ \forall\, \xi_i \sim \text{Pois}(\lambda), \xi_i\text{ are independent } \implies P\left(\limsup\limits_{n \rightarrow \infty}\frac{\...
0
votes
1answer
17 views

If a linear objective function over a constriant can attain a finite maximum, then it must be able to obtain this maximum on a boundary point

Let $C\subset \Bbb R^n $ be a non-empty constraint (may be non-convex) and $f({\mathbf x}) = {\mathbf w}^{\operatorname{T}}{\mathbf x}:C \to \Bbb R$ be a linear real-valued objective function. If $f$ ...
0
votes
1answer
33 views

Let $g_n(x)=\sum_{k=1}^n (-1)^k f_k(x) \forall x\in \mathbb R. $ Then which one of the following are correct answers?

Suppose that $\{f_n\}$ is a sequence of continuous real-valued functions on $[0,1]$ satisfying the following: (A)$\forall x\in \mathbb R,\{f_n(x)\}$ is a decreasing sequence. (B)the sequence $\{f_n\}...
0
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0answers
58 views

about Lebesgue measure on R

E is a set such that $m(E)>0$, $E \subset (0,1)$ and there exist $c>0$ such that for some moving interval $I$, $$\lim_{mI\rightarrow0}\frac{m(E\cap I)}{m(I)}=c$$ Proof:mE=1 My attempt ,I have ...
-1
votes
0answers
40 views

Explicit calculation of the given integral

Can we calculate this integral explicitly? $$\huge\int\frac{dx}{1+e^{-k\left(\sin^2\left(\frac{\Gamma(x)+p}{x+q}\right) - 1\right)}}$$
0
votes
1answer
16 views

Correct Proof of Uniform Convergence?

Let $f_{n}(x) = \int_{0}^{x}\min\{n, \frac{1}{\sqrt{t}}\} dt$ and $f(x) = \int_{0}^{x}\frac{1}{\sqrt{t}}dt$. Show $f_n(x) \overset{n\to\infty}{\longrightarrow} f(x)$ uniformly. Current attempt: \...
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0answers
18 views

Riemann Integrable definition in partition distingued $\mathbb{Q}$

A function $f:[a,b]\rightarrow \mathbb{R}$ is Riemann Integrable on $[a,b]$ if $\exists \thinspace L \in \mathbb{R} : \forall \epsilon >0\thinspace \exists \thinspace \delta >0 :$ if $P$ is any ...
2
votes
1answer
95 views

If $\int_0^1 f(x)x^n \ dx=0$ for odd natural numbers, then $f\equiv 0$.

$f(x)$ is continuous. I know the proof when this is true for all $n$. But I don't know how to proceed when it is only talking about odd $n$'s.
1
vote
2answers
46 views

Can I say $1/x^2$ is not uniformly continuous on $[0, \infty)$ because it is not defined when x = $0$?

This is in my homework to show if $1/x^2$ is uniformly continuous on $[0, \infty)$. I'm thinking since the definition of uniform continuity says for all $ x, y \in $ the domain, $|f(x) - f(y)| < \...
1
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0answers
22 views

Is the function $f(x) = \sum_{n\geq 1 } \cos(n^2 x ) / n^{2.6}$ absolutely continuous?

It is of course continuous. Here the exponent $2.6 $ is delibrately chosen to be bigger than $2.5$. If we differentiate the series formally term-by-term, we will get $$ - \sum_n \frac{\sin n^2 ...
1
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0answers
36 views

How many times $f-g$ changes sign in $(0,\infty)$

Consider $f(x)=e^{-x^2}$ and $g(x)=\frac{1}{1+x^2}$ Question: How many times $f-g$ changes sign in $(0,\infty)$ Let us call $h=f-g$ Note that $e^{x^2}=1+x^2+x^4/2!+\dots \geq 1+x^2$ so $e^{-x^2}&...
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0answers
21 views

Prove Jacobian of $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ with 3 conditionals over $\mathbb{R}^{2}$ is $I_{2 \times 2}$.

If $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is given by: $f(x,y)= \begin{cases} (x,y-x^{2}) & if & x^{2} \leq y \\ (x,\frac{y^{2}-x^{2}}{x^{2}}) & if & 0 \leq y \leq x^{...
0
votes
0answers
31 views

What is the $\lim_{t\to \infty} u(1,t)$

Let $u(x,t)$ be a solution of $u_t-u_{xx}=0$ we are given the following additional information. $$u(x,0)=\frac{e^{2x}-1}{e^{2x}+1}$$ and $u(x,t)$ is bounded. I want to find the $\lim_{t\to \infty} ...
0
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0answers
23 views

Real zero of a positive real function

I have a real positive function defined over very large positive real domain which has a single zero in that domain. Function is infinitely differentiable. I am looking for an algorithm to find this ...
7
votes
2answers
103 views

What's wrong with this proof that $0 = 1$?

Let $$f_n(x)=\frac{1}{\sqrt{\pi n}}e^{-x^2/n}.$$ Note that $f_n(x)\to 0$ uniformly as $n\to\infty$. [Proof: $0\leq f_n(x)\leq\frac{1}{\sqrt{\pi n}}$; given any $\epsilon > 0$, let $M=\left\lceil\...
1
vote
1answer
18 views

If $0 = \left(TH\right)^\perp = \ker T^*$, then $TH = H$.

If $H$ is a Hilbert space and $T$ is a bounded linear operator on $H$ with the following property $$0 = \left(TH\right)^\perp = \ker T^*$$ where $T^*$ is injective, then $TH = H$. Discussion I've ...
0
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0answers
23 views

Real Analysis: Functions, Continuity, modulus [on hold]

Let $f : \mathbb{R} → \mathbb{R}$ be a function. Suppose that there exists $K > 0$ such that for all $x, y ∈$ $\mathbb{R}$, we have $|f(x) − f(y)| ≤ K|x − y|$. Show that f is continuous.
1
vote
1answer
19 views

every infinite subset of a metric space has limit point => metric space compact?

I am self-studying the book, Walter Rudin, Principles of Mathematical Analysis, and I am doing the Exercise 2.26. As mentioned in the hint, Exercise 23 suggests: every separable metric space has a ...
0
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0answers
9 views

Finding a set that satisfies this property

i am looking for a set that satisfies this property but im lost , can you help me out ? $\underline{c}(E) < m^\star(E) < \overline{c}(E)$ where this $c$ is referring to the jordan content. I ...
2
votes
1answer
43 views

Function is Riemann integrable in $[a,c]$ and $[c,b]$ then is RI in $[a,b]$

i need to prove that if Riemann integrable in $[a,c]$ and $[c,b]$ then is RI in $[a,b]$, maybe is easy but i can't see it. Definition of Riemann Integral A function $f:[a,b]\rightarrow \mathbb{R}...
2
votes
1answer
64 views

Inequality for a differentiable concave function on $[0,1]$

Note: Concave refers to this definition. Let $f: [0,1] \rightarrow \mathbb{R}$ be differentiable, concave and not identically $0$. Assume $f(0) = f(1) = 0.$ Let $a \in (0,\frac{1}{2})$. Show that $$...
5
votes
1answer
72 views

Request for info about Real Analysis to a beginner [on hold]

So I'm a junior HS student very interested in mathematics and I think I have pretty good chances to get admitted into Stanford online HS and take a university level real analysis course that's worth 5 ...
0
votes
0answers
21 views

Is $f \in L^2(M)$?

Let $$M=\lbrace rcos(\phi),rsin(\phi)\in \mathbb{R^2}:0<r<1,-\frac{\pi}{2}<\phi<\frac{\pi}{2}\rbrace$$ and $$ f(rcos(\phi),rsin(\phi))=rcos(\phi)$$ I have to show that $$ \int_M|f|^2d\mu &...
0
votes
0answers
12 views

Showing $\sup_{\Vert w \Vert_\infty} |v^\ast w| = \Vert v \Vert_1$ for $v \neq 0$

Problem Show, for $v \neq 0 \in \mathbb{C}^n$ $$ \sup_{\Vert w \Vert_\infty=1} |v^\ast w| = \Vert v \Vert_1 $$ And find the similar equality for $\sup_{\Vert w \Vert_1=1} |v^\ast w|$ Try ...