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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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3answers
74 views

Proving a function $\frac{1}{2y}\int_{x-y}^{x+y} f(t) dt=f(x)$ is a linear polynomial

Here's the question: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $$\frac{1}{2y}\int_{x-y}^{x+y} f(t) dt=f(x)\text{ for all }x\in\mathbb{R},y>0$$ Show that ...
2
votes
2answers
83 views

Baby Rudin Theorem 2.33

Theorem 2.33 Suppose $K \subset Y \subset X$. Then $K$ is compact relative to $X$ if and only if $K$ is compact relative to $Y$. I understand the first part of the proof: if $K$ is compact relative ...
2
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1answer
36 views

let $a_{k}=\frac{k^4-17k^2+16}{k^4-8k^2+16}$ real number for $5≤k\in N$

If $a_{k}=\frac{k^4-17k^2+16}{k^4-8k^2+16}$ real number for $5≤k\in N$ Then find : $\lim_{n\to +\infty} a_{5}a_{6}a_{7}...a_{n}$ My try : $k^4-17k^2+16=(k^2-1)(k^2-16)$ And $k^4-8k^2+16=(k-...
0
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1answer
10 views

The quantities x and y w that the relativeusing x√y to approximate x√y is given by . |∆x/x|+1/2|∆y/y|

The quantities x and y has been estimated using X and Y with errors ∆xAnd ∆y respectively . show that the relative error in using x√y to approximate x√y is given by . ...
2
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0answers
10 views

Arbitrary signed and absolutely continuous measure has a derivative w.r.t a positive measure

This is an exercise from Folland Real Analysis p.92 that I am just totally stuck at... It was not hard to prove b. and c. under the assumption of a. But then, I cannot do anything (literally anything)...
3
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0answers
21 views

Harmonic extension for harmonic function

Studying the mean spherical mean and the volumetric mean, this question has occurred to me. The volumetric mean is defined as follows: Let $\Omega \subset \mathbb{R}^n$ open and $f:\Omega \to \...
1
vote
1answer
60 views

How to solve an indefinite integral using the Taylor series?

I am trying to show that the following integral is convergent but not absolutely. $$\int_0^\infty\frac{\sin x}{x}dx.$$ My attempt: I first obtained the taylor series of $\int_0^x\frac{sin x}{x}...
12
votes
3answers
222 views

Covering a compact set with balls whose centers do not belong to other balls.

Let $K\subset \Bbb R^n$ be a compact set such that each $x\in K$ is associated with a positive number $r_x>0$. Claim: $K$ can be covered by a family of balls $$ \mathcal B = \{ B(x_i,r_i) : i=...
0
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2answers
38 views

If $f$ is lipschitz, then $|f(x)| < C \left(1+|x|^λ\right)$

I'm in a first course of analysis and we got this question and I wasn't able to figure it out. Any hint's are welcome. If $f$ is lipschitz, then $\vert f(x)\vert<C(1+\vert x\vert ^λ)$ for some $C,\...
3
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1answer
54 views

Is this function continuous at x=2?

I have a function, k, below that I think is not continuous at x=2, but I'm not sure. If it is how can it be proven. Let: $h(x) = \frac{x^2+20}{6} $ $g(x) =\frac{12+8x-x^2}{6}$ $t(x) =4+\frac{2}{3}(...
2
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0answers
24 views

Taylor expansion and the intermediate value theorem

Suppose $f : \mathbb R^2 \to \mathbb R$ is a differentiable function at a point $\mathbf a =(a,b) \in \mathbb R^2$. Then for $\mathbf x = (x,y) \in \mathbb R^2$ close to $\mathbf a$, Taylor's theorem ...
2
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0answers
19 views

Show that the borel $\sigma$-algebra on $M$ is the unique $\sigma$-algebra whos restriction to each $U_\alpha$ is the pullback via $\pi_\alpha$

Let $M$ be an n-dimensional manifold with atlas $\pi_{\alpha}: U_{\alpha} \rightarrow V_{\alpha}$. Show that the borel $\sigma$-algebra on $M$ is the unique $\sigma$-algebra whos restriction to each $...
2
votes
2answers
77 views

If partial derivatives of a harmonic function are constant, is the function linear?

Let $u: \mathbb{R}^2 \to \mathbb{R}$ be a harmonic function. If $\frac{\partial u(x,y)}{\partial x} = k_1$ and $\frac{\partial u(x,y)}{\partial y} = k_2,\forall (x,y) \in \mathbb{R}^2, k_1, k_2 \in \...
0
votes
1answer
35 views

Pointwise convergence vs convergence in measure [on hold]

I would be very grateful if I could be given an example of a sequence convergent pointwise but it is not convergent in measure and an example of a sequence convergent in measure but it is not ...
3
votes
1answer
64 views

How many inner products exist in $R^n$?

I want to know how many inner products exist in $R^n$ making it a Hilbert space. I know they all can be corresponded to the Euclidean inner product by some Isometry/Unitary function, but I want to ...
2
votes
4answers
70 views

The integral of some function of $[x]$

I am trying to take the following integral which sounds easy but I really got confused and need help. $f: [0,2]\to \mathbb{R}$ is integrable. Show that $$\int_0^2(x-1)f[(x-1)^2]dx=0.$$ For me the ...
1
vote
2answers
30 views

monotonic function can only have simple discontinuity

I am self-studying Rudin, Principles of Mathematical Analysis. I am having trouble going through the theorem saying that monotonical functions can only have simple discontinuity, i.e., Suppose $f$ is ...
0
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2answers
38 views

Find for which lambda values this equation gives positive solutions [on hold]

The equation is $x^3 +x^2 +3 = \lambda x$ I thought I could move lambda to the first sector, getting $x^3 +x^2 -\lambda x+3 = 0$ But then? Can you show me the solution?
0
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1answer
45 views

Extended Real Plane - Natural generalization of the two point compactification of $\mathbb R$

Is the extended $\overline{\mathbb R}^2$ written as $\mathbb R^2 \cup \{-\infty,+\infty\}$ or $(\mathbb R\cup \{-\infty,+\infty\})^2$ ? What is the name for the latter set? A.k.a Are there four ...
0
votes
1answer
31 views

I have tried answering this but can't get the right answer [on hold]

Consider three circles all tangent inscribed in a rectangle find the area of the rectangle in terms of the radius of the circles
0
votes
1answer
26 views

Check differentiability of $(x,y)$ at $(0,0)$.

Check differentiability of $f(x,y)$ at $(0,0)$ : $f(x,y)= \frac {xy} {\sqrt {x^2 +y^2}}$ when $(x,y)\neq 0$ and $0$ when $(x,y) = 0$. What definition shall I use?
0
votes
0answers
15 views

minor question about relative openness

if 𝐴 is open relative to 𝐵, does this mean that 𝐴 is a subset of 𝐵? I guess yes, right?
0
votes
0answers
37 views

The maximum of roots

There are 10 such square trinomials $ P_1, P_2, ..., P_ {10} $, such that their graphs touch each other in pairs. How many maximum different roots can a polynomial have $(P_1-P_2) (P_2-P_3)...(P_9-P_ {...
1
vote
1answer
20 views

How To Analyze Statements of Quantifiers

Having trouble figuring out how to interpret Universal Quantifiers, from my book there's two sets of statements. Assuming x,y and z are real numbers, determine the truth value of each statement (a): $...
1
vote
4answers
72 views

Does the following limit exist $\lim_{(x,y)\rightarrow (0,0)}\frac{\sin(xy)}{y}$

Does the following limit exist? $\lim_{(x,y)\rightarrow (0,0)}\frac{\sin(xy)}{y}$ The answer here is not correct in my opinion since it does not take under considerarion all the surface parts in ...
1
vote
1answer
38 views

Does $f_n\to0$ pointwise + $f_n$ integrable + $f_n$ uniformly bounded imply $\int f_n\to 0$?

If $\{f_n:[0,1]\to\mathbb{R}\}_{n\ge 1}$ is a sequence of Riemann-integrable functions that converge pointwise to the zero function and $\{f_n\}_{n\ge 1}$ is uniformly bounded by $M$ can we prove that ...
0
votes
1answer
26 views

Finding some points with which one can calculate the integral of multiplication of two functions [duplicate]

Let $f,g:[a,b]\to\mathbb{R}$ be two Riemann integrable functions. For each partition $P=\{t_0,...,t_n\}$ of $[a,b]$ take some points in two different ways, that is by choosing two points $\eta_i$ and $...
1
vote
1answer
49 views

Rudin, theorem 2.30 intuition behind

could you show 1-2 real examples that is related to this theorem? 2.30 Theorem: Suppose 𝑌⊂𝑋. A subset 𝐸 of 𝑌 is open relative to 𝑌 if and only if 𝐸=𝑌∩𝐺 for some open subset 𝐺 of 𝑋.
1
vote
1answer
27 views

Pattern in Squared Numbers and their Digit Sum

So this has been boggling my mind for some time now. On spring break I was really bored and started messing around with numbers when I noticed something. I was squaring each number (1-9) when I ...
0
votes
3answers
33 views

Prove that $0< \frac{1}{2^{m}} <y$

If $y$ be a positive real number, show that there exists a natural number $m$ such that $0< \frac{1}{2^{m}} <y$ I think I have to use Archimedean property to prove it. The Archimedean property ...
1
vote
1answer
38 views

Showing that a harmonic function is constant

Let $u:\mathbb{R}^n \to \mathbb{R}$ be harmonic function and suppose that exist $C>0$ such that $|u(x)| \leq C(1+\sqrt{\|x\|})$. I want show that $u$ is constant. My first idea: Show that $C(1+\...
0
votes
1answer
28 views

Is there any example which can show that the change of the order of integration is not plausible when the improper integral exists?

Let $f(x,y)$ be continuous on $Q=\{(x,y)\mid x>0,y>0\}$ and $\iint_Q |f(x,y)|$ converge. $$\int_{0}^{\infty}\int_{0}^{\infty}f(x,y)\, \mathrm{d}x\, \mathrm{d}y \stackrel{?}{=}\int_{0}^{\infty}\...
-1
votes
0answers
34 views

Homogeneous inequality with three variables [on hold]

Let $a,b,c>0$ then we have : $$\sqrt{\frac{abc}{a+b+c}}\Big(\sum_{cyc}\frac{1}{7a+b}\Big)\leq \sqrt{\frac{\sqrt{abc}}{\sqrt{a}+\sqrt{b}+\sqrt{c}}}\Big(\sum_{cyc}\frac{1}{7\sqrt{a}+\sqrt{b}}\Big)$$...
0
votes
1answer
57 views

Inequality $\frac{abc}{\sqrt{3}\sqrt{a^2+b^2+c^2}}\Big(\sum_{cyc}\frac{1}{7a^2+b^2}\Big)\leq \frac{1}{a+b+c}\Big(\sum_{cyc}\frac{a^3}{7a^2+b^2}\Big)$

Let $a,b,c>0$ then we have : $$\frac{abc}{\sqrt{3}\sqrt{a^2+b^2+c^2}}\Big(\sum_{cyc}\frac{1}{7a^2+b^2}\Big)\leq \frac{1}{a+b+c}\Big(\sum_{cyc}\frac{a^3}{7a^2+b^2}\Big)$$ My try : Here ...
3
votes
1answer
62 views

Mean value theorem for a $C^1$ function

Let $f:\mathbb{R^+} \to \mathbb{R}$ be $C^1$ and let's define for all $x>0$ $$c(x)=\inf \left\{ (f')^{-1}\left(\left\{\frac{f(x)-f(0)}{x}\right\}\right) \bigcap \,[0,x]\right\}$$ i.e $c(x)$ is ...
2
votes
2answers
79 views

Showing that $f(x)=x^3-3x+1$ has at least two zeros in the interval $[0,2]$

I was given this task by my professor: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be defined by $f(x)=x^3-3x+1$ Show that $f$ has at least two zeros in the interval $I := [0,2]$. My answer is: Since ...
1
vote
0answers
22 views

Is the dominated convergence theorem applicable whenever “THIS” theoem is applicable?

THIS theorem: Let $I =[a,b]$ be a closed and bounded interval and $\forall n\in \mathbb{N}$, $f_n:I \to \mathbb{R}$ be Riemann integrable on $I$. If the sequence $(f_n)$ converges uniformly to a ...
1
vote
0answers
33 views

Taylor's series in the binomial approximation to Poisson random variable.

The Poisson random variable has a probability distribution $$P_X(k) = e^{-(\lambda T)} \frac{{(\lambda T)}^k}{k!}$$ We can relate this expression to the binomial distribution by dividing the temporal ...
0
votes
0answers
28 views

Decreasing sets in complete metric spaces

Banach gave a proof in 1920 of: in complete normed vector spaces, the sequence of the centres of decreasing closed balls has a limit which is in every ball. Any idea of a previous proof of such ...
6
votes
0answers
56 views

If $f_n \rightarrow 0$ with $f_n ' \rightarrow g$, then is $g=0$ in some sense?

Suppose $f_n :[a,b] \rightarrow \mathbb{R}$ are differentiable functions (need not be $C^1$) with $f_n \rightarrow 0$, $f_n ' \rightarrow g$ pointwise. Can we say that $g=0$ in some sense? (Say, a.e.)...
-1
votes
1answer
29 views

Existence of a hyper plane. [on hold]

Is it possible to use Hahn-Banach Theorem to solve this problem. If so please give a precise solution.
1
vote
0answers
56 views

Evaluate $\lim_{n \to \infty}\int_{0}^{1}\frac{n+\cos^n(e^x)}{4n+x^4} dx$

Evaluate $\lim_{n \to \infty}\int_{0}^{1}\frac{n+\cos^n(e^x)}{4n+x^4} dx$ Attempt: We define $f_n(x)=\frac{n+\cos^n(e^x)}{4n+x^4}$ on the domain $(0,1)$. This sequence of functions $(f_n)$ converges ...
0
votes
1answer
32 views

let $f: J \to \mathbb{R}$ . let $x_n < c <y_n$ be such that $(y_n-x_n) \to0$ show that $\lim _ {n \to \infty} \frac{f(y_n)-f(x_n)}{y_n-x_n}=f'(c)$ [duplicate]

let $f: J \to \mathbb{R}$ be differential function . let $x_n < c <y_n$ be such that $(y_n-x_n) \to0$ show that $$\lim _ {n \to \infty} \frac{f(y_n)-f(x_n)}{y_n-x_n}=f'(c)$$ i am trying to ...
4
votes
1answer
41 views

Equivalency of all norms on a finite dimensional vector space: compactness theorems vs. the open mapping theorem

Going through my functional analysis course notes, I feel like there are two different proofs for the following theorem. In $\mathbb{R}^n$ (or $\mathbb{C}^n$), any two norms are equivalent. One ...
-1
votes
2answers
28 views

Prove that f(x) = absolute value of x when x is not 0 and f(0)=9 on the closed interval [-1, 1] is integrable. [on hold]

I must use the Riemann Integral somehow in this proof, but I am completely lost and have no clue where to begin. Prove that f(x) = absolute value of x when x is not 0 and f(0)=9 on the closed ...
1
vote
1answer
27 views

Seemingly faulty proof of the Integrability Criterion

I may have discovered a quick way of proving the following Theorem, but the proof is quite short and hence `too good to be true.' Could you please take a look at it and see if it makes sense? THEOREM ...
0
votes
0answers
54 views

An example of a non integrable function which has antiderivative.

I am looking for a non Riemannian integrable function which has primitive. Any idea if such an example exists?
0
votes
0answers
47 views

If $f$ be a uniformly continuous function on $(a,b)$ then $f$ is bounded there.

We define a continuous extension of $f(x)$ to the set $[a,b]$, by $g(x)=f(x), x\in (a,b)$ and $g(a)= \lim_{x \to a^{+}}f(x)$ and $g(b)=\lim_{x \to b^{-}}f(x)$. $g(x)$ being continuous on a compact set ...
2
votes
0answers
42 views

Games and Dyadic Rationals

Suppose $a$, $b$, and $c$ are dyadic rationals, where their corresponding numbers (games) are $A$, $B$, and $C$ respectively. Prove that $a+b=c$ if and only if $A+B\sim C$ (or $A+B$ is equivalent to $...
0
votes
1answer
52 views

Limit of $\frac{2^{n^k}}{2^{2^n}}$

$\newcommand\iddots{\mathinner{ \kern1mu\raise1pt{.} \kern2mu\raise4pt{.} \kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.} \kern1mu }}$What is the limit as $n$ tends to infinity of $$\frac{2^{n^k}}{...