Theoretical foundations of calculus: 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
9 views

Proofs involving the Least Upper and Greatest Lower Bound Properties

Please help me with the following question: Let $S \subset T \subset \Bbb{R}$, then if $T$ is bounded from above, then $\sup S \le \sup T$ if $T$ is bounded from below, then $\inf T \le ...
3
votes
1answer
23 views

Convolution of two functions is a constant

I have the convolution of two functions $f(t)$ and $g(t)$ is a constant $K \in \mathbb{R}$, i.e., $(f*g)(t) = K$ for all $t \in \mathbb{R}$. Taking Fourier transform of the convolution yields $$ ...
3
votes
3answers
68 views

$\ln(1+x)$ — why does its power series converge for $|x| < 1$?

Is it a similar reasoning for the convergence of a geometric series, when $|x|<1$? ...at $-1$, $\log(1+x) = \log(0)$, which is undefined. For $x < -1$, $\log(1+x)$ is negative, which is also ...
5
votes
1answer
53 views

Can a $1d$ space never be curved?

I was wondering about this: Wikipedia article I refer to (here I refer to the first part: metric) This wikipedia article claims that this hyperbolic space model has constant curvature $-1.$ I believe ...
1
vote
1answer
44 views

Find a function with the property, or prove it doesn't exist

Today, I encountered the following problem in my research. I'd like to find a function $f(x_1, x_2, \ldots, x_n)$ such that $$ 0 = \frac{d f(x_1, x_2, \ldots, x_n)}{d a}\bigg|_{c_1,c_2,\ldots,c_n} ...
1
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1answer
27 views

How can I construct such a Borel subset?

This is a problem from the book "Real analysis for graduate students" : Let $m$ be a Lebesgue measure. Construct a Borel subset $A$ of $\mathbb{R}$ such that $0<m(A\cap I)<m(I)$ for every open ...
6
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3answers
53 views

Prove that $|\sin^{−1}(a)−\sin^{−1}(b)|≥|a−b|$

Question: Using the Mean Value Theorem, prove that $$|\sin^{−1}(a)−\sin^{−1}(b)|≥|a−b|$$ for all $a,b∈(1/2,1)$. Here, $\sin^{−1}$ denotes the inverse of the sine function. Attempt: I think I know ...
2
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1answer
25 views

Multidimensional “bump” function for any closed subset

I have a function $f\in C^\infty: \mathbb{R}^n \to \mathbb{R}$ with the properties : 1) $f(x) = 0$, when $\Vert{x}\Vert \le 1$, 2) $0 < f(x) < 1$, when $1 < \Vert{x}\Vert < 2$, 3) $f(x) ...
-3
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1answer
45 views

can anyone help me with following question attached in image file

Let $(X,\|\cdot\|)$ be a normed space, where $$X=\{(a_n)_{n\geq 1} \mid (a_n)_{n\geq 1} \text{, bounded real sequence}\}$$ and $$\|(a_n)_n\|=\sup_{n\in N} |a_n|$$ Let $$ M=\{(a_n)_n\in X\mid 0\leq ...
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0answers
20 views

Prove that norm of Yosida approximations blows up

I need an help with the following exercise about Yosida approximations. Let's fix the notation: let $A$ be an unbounded linear operator define on a dense domain $D(A)\subset X$ of a Banach space $X$. ...
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0answers
19 views

Let $\{x_n\}$ be a sequence of positive reals, show that $\limsup\sqrt[n] {x_n}\leq\limsup \frac{x_{n+1}}{x_n}$.

Let $\{x_n\}$ be a sequence of positive reals, show that $\limsup\sqrt[n] {x_n}\leq\limsup \frac{x_{n+1}}{x_n}$. This come from a problem set, in which $\limsup{\sum^{n}_{i=1}\frac ...
1
vote
1answer
13 views

$f$ is bounded by $M$ on $[a, b]$ and if the restriction of $f$ to every interval $[c, b]$ where $c$ in $(a, b)$ is Riemann integrable

If $f$ is bounded by $M$ on $[a, b]$ and if the restriction of $f$ to every interval $[c, b]$ where $c$ in $(a, b)$ is Riemann integrable, then $f$ is Riemann integrable and that $\int _c^b f \to ...
2
votes
1answer
26 views

How do I complete this proof that the absolute value of an integral function is an integrable function?

I'm trying to complete the proof in this answer that if $f: [a, b] \to \mathbb{R}$ is a Riemann integrable function, then $|f|$ is an integrable function also. I understand the proof that $$ ...
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3answers
30 views

Uniform convergence epsilon delta

How can I show that the function $ f_n(x)=\frac{nx}{1+nx} $ converges uniformly on the interval $ x \in [1, \infty ) $ I have already proven the pointwise limit to be $ f(x)=1 $ I am working with ...
1
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1answer
18 views

Is there an analytical solution for a single unknown in this summation?

I have the following relation: $$ y = \sum_{i=1}^N \frac{f_i}{1+e^{g_i + \alpha}},$$ where $N\in\mathbb{N}$, $y,\alpha, f_i,g_i\in\mathbb{R}$ and $\sum_{i=1}^{N}f_i=1$. Furthermore, all parameters ...
2
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2answers
20 views

$f$ is Riemann integrable

We know that if $f \in \mathcal R[a,b]$ and if $a = c_0 < c_1<\cdots<c_m =b$, then the restrictions of $f$ to each subinterval $[c_{i-1},c_i]$ are Riemann integrable. Is the converse true, ...
0
votes
1answer
9 views

What is the mean value theorem for the Fréchet (total) derivative?

What is the mean value theorem for the Fréchet (total) derivative? Off the top of my head, it's something like $$ \|F(x+h)-F(x)\|\leq \sup_{c\in[0,1]} \|F^\prime(x+ch)\|\|h\| $$ but the double ...
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2answers
34 views

Could someone explain me the task?

this is the question: Show that for each linear map $f:\mathbb R^d → \mathbb R^e$ there exists $a < \infty$ so that $\|fw\|< a\|w\|$ for each $w$ in $\mathbb R^d.$ And my problem is that $f$ ...
0
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2answers
25 views

How do we show that limit $\frac{x^6-x^2sin(\frac{1}{x^2})}{x^4}$ as x tends to $0$ does not exist?

How do we show that the limit of $\frac{x^6-x^2sin(\frac{1}{x^2})}{x^4}$ as x tends to $0$ does not exist? I thought maybe we should consider two sequences that tend to 0 and show than $f(a_{n})$ and ...
3
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1answer
24 views

Convergence for every measurable set

Let $(f_n)$ non-negative measurable functions such that $f_n\to f$ and $\int f_n\to \int f<\infty$. We have to prove that $\int_E f_n\to \int_Ef$ for each $E$ measurable. I know that if $f_n\to f$ ...
0
votes
1answer
38 views

some properties of $\nu$ measure

For any given function $F$ satisfying the following properties $0\le F(x)\le1,\forall x\in\mathbb R$ $F(x)\le F(y),x\le y$ $\lim_{x\to-\infty}F(x)=0,\lim_{x\to\infty}F(x)=1$ $F$ is continuous from ...
2
votes
2answers
31 views

nonnegative Riemann-integrable function, infimum

$f$ is a nonnegative Riemann-integrable function on $(0,1)$ and $f(x)\ge\sqrt{\int_0^xf(t)dt}$ for $x\in(0,1)$. Find $\inf\frac{f(x)}{x}$ I have no idea how to work out the assumption, for equality ...
1
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2answers
28 views

Determine interior and boundary of a set

Let $(X,||\cdot||)$ be a normed vector space, where $$X = \big\{ (a_n)_{n \geq 1} ~~|~~ (a_n)_{n \geq 1} \text{ is a bounded real sequence }\big\}$$ and $$\|(a_n)_n\| = \sup_{n \in ...
0
votes
0answers
15 views

Jacobi field along every geodesic?

I stumbled over the question: Given a manifold $M$. Can there exist a vector field that is a Jacobi field along every geodesic? Now, the answer is apparently: Yes, because the $0$ vector field does ...
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0answers
15 views

convexity of a function of 2 variables

$f\colon\mathbb{R}\to\mathbb{R}$ is continuous and $|f|$ is convex. Prove that $F\colon\mathbb{R}^2\to\mathbb{R}$ defined as $F(x,y)=|f(x)|+|y-f(x)|$ is convex.
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0answers
18 views

Normal coordinates and the metric tensor

I was wondering whether the metric tensor in normal coordinates can be expressed somehow in terms of the exponential map. Cause I just don't see how the metric in normal coordinates is actually ...
1
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2answers
51 views

How to find the limit points of the set $\{\ a+\alpha\ b \mid a,b \in \mathbb Z, \ \alpha\ \text{is a fixed irrational number} \}$

How to find the limit points of the set $\{\ a+\alpha\ b \mid a,b \in \mathbb Z, \ \alpha\ \text{is a fixed irrational number} \}$ limit point: A point $x$ is said to be a limit point of a non empty ...
1
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0answers
18 views

Suppose that $P$ is a monic polynomial of degree $n$ in one variable with real coefficients and $K$ a real number.Then which is /are true?

Suppose that $P$ is a monic polynomial of degree $n$ in one variable with real coefficients and $K$ a real number.Then which is /are true? If $n$ is even and $K>0$ , then there exists $x_0 \in ...
1
vote
1answer
13 views

eigenvector of a multiplication operator

Let $\phi\in L^\infty(\mu)$. Define $M_\phi:L^2(\mu)\to L^2(\mu)$ by $M_\phi(f)=\phi f$. What conditions are needed on $\phi$ so that $M_\phi$ has an eigenvector? If $\phi$ is a constant clearly ...
8
votes
1answer
95 views

Does equation $f(g(x))=a f(x)$ have a solution?

Suppose $g: [0,1] \to [0,1]$ is a strictly increasing, continuously differentiable function with $g(0)=0$ and $g(1) < 1$, and $a \in (0,1)$. Is there a function $f:[0,1] \to \mathbb{R}$ such that ...
3
votes
3answers
58 views

a linear differential equation with periodic coefficients

Let $$y' = a(x) y + b(x)$$ be a linear differential equation with continuous, periodic coefficients $a, b: \mathbb{R} \to \mathbb{R}$ that both have a period of $T > 0$. Also, we assume that ...
5
votes
5answers
58 views

Show $\sum n e^{-na}$ converges for $a>0$

Is there any test or property in particular I can use to show $ \sum n e^{-n a}$ is convergent for $a>0$ ? I think it is obvious that from looking at the function that this is convergent, since ...
0
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0answers
18 views

follow-up question to Hake's theorem in Bartle's book

My question is based in here. Why is it that $b$ forces to be a tag of $[x_{m-1},b]$? I can't get the right trick. Can you please give me some hints? Thanks
2
votes
1answer
46 views

Showing Uniform convergence of $\frac{n x}{1 + n \sin(x)}$

I want to prove for all $a\in \left(0,\frac{\pi}{2}\right]$, $ \ f_n\to f$ uniformly on $\left[a,\frac{\pi}{2}\right]$. Also, how is this different from $f_n \to f$ uniformly on $\left(0, ...
0
votes
0answers
30 views

Proving that region bounded by y=0 and continuous function is a Jordan region.

How do I prove the following? Let $f(x)$ be continuous on $[a,b]$ and let $A=\{(x,y): x \in [a,b] \text{ and } 0 \le y \le f(x)\}$. Prove that $A$ is a Jordan region. I know that I can show ...
0
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1answer
34 views

Uniform Convergence of $(\frac{x}{1+x})^n$

I have an exercise based on the exercise 8 of page 41 in Complex Analysis of Ahlfors. In that exercise they ask for the values of x in which the following series converges: $$ ...
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2answers
60 views

Find all continuous real valued function such that $(f(x))^2+C=\int\limits_0^xf(t)dt$

Find all continuous real valued function such that $$(f(x))^2+C=\int\limits_0^xf(t)dt$$ for some $C\in\mathbb{R}$ If I set $F(x)=\int\limits_0^xf(t)dt$ then $F$ is differentiable and ...
0
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0answers
15 views

Upper contingent derivative of a lsc function

Let $F : \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ be a set-valued map, locally bounded, upper semi-continuous, and taking nonempty, convex and compact values. Let $\,f : \mathbb{R}^n \to ...
2
votes
1answer
27 views

Radon-Nikodem Derivative of a purely nonatomic Borel Measure

If $\mu$ is a purely non-atomic Borel measure on a topological space $X$ then must its density be a continous function to $\mathbb{R}$? My intuition says yes because all my counterexamples are not ...
2
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2answers
46 views

$\mathrm{d}f(x,t)$ this way $d\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} \,dt+\frac{\partial f}{\partial x}\,dx$?

If $dX_t=a_t \,dt$ the next procedure is correct? $$\mathrm{d}\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} dt+\frac{\partial f}{\partial x}dx=\frac{\partial f}{\partial t} dt+\frac{\partial ...
2
votes
0answers
55 views

Finding the maximum of two functions with complicated formulas

Let $$ f(\omega)=1+\frac{m(a+\omega^2)}{a^2+\omega^2}+\alpha\left(\frac{a^2+\omega^2-ma}{a^2+\omega^2}\right)\cos(\omega\tau)+\frac{\alpha m\omega}{a^2+\omega^2}\sin(\omega\tau)\;, $$ and $$ ...
1
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0answers
20 views

finding Interior Points of a set

In the normed space $(\mathbb{R}^2, ||(x_1,x_2)||:=|x_1|+|x_2|)$ I want to find Interior Points of $$ \{ (x,1/n) ~~\big|~~ x\in \mathbb{R} \text{ and } n\in \mathbb{N} \}. $$ I guess that the ...
-3
votes
0answers
30 views

Change the subject of a formula [on hold]

$150 \cdot 10^6 = \dfrac{3pR^2}{4t^2}$ How do I find out what $t$ is, hence make it the subject of the equation. I think I know what the answer should be: $p=1.5 \cdot 10^6$ $R= 0.075$ ...
2
votes
3answers
52 views

Polynomial of degree $2$ has at most $2$ roots

Suppose that $P: \mathbb{R} \rightarrow \mathbb{R}$ is a real polynomial of degree exactly $2$. Prove $P$ has at most two roots. Let $P(x)=a_2 x^2 +a_1 x +a_0$ for all $x \in \mathbb{R}$. I tried to ...
0
votes
1answer
12 views

How I find a suitable increment value

If I have three variables $x,y$ and $z$, where $x\lt z$ and $y\lt z$, then I need to make each value of $x$ and $y$ equal to or approximately equal to $z$ by adding a ratio of another variable, for ...
3
votes
2answers
89 views

Does $\frac{nx}{1+n \sin(x)}$ converge uniformly on $[a,\pi/2]$ for all $a \in (0,\pi/2]$?

Edit: the question had some missing details. It should read as follows: Prove for all $a \in (0,\frac{\pi}{2}]$, $f_n \rightarrow f$ uniformly on $[a,\frac{\pi}{2}]$. Here $$f_n(x) = \frac{n x}{1 ...
0
votes
1answer
34 views

Bounded sequence of positive numbers

Suppose that $\{x_n\}$ is a sequence of positive real numbers so that $\lim \frac{x_{n+1}}{x_n}=L<1$. Then show that for $n$ large enough and for some $C>0$ we have $0<x_n<Cr^n$. I have ...
0
votes
2answers
40 views

How prove this inequaliy $\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt $

let $f \in C^1([a, b])$ with $a, b \in \mathbb{R}, a < b$ show that $$\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt$$ I've tried to use the ...
0
votes
1answer
40 views

Real Analysis Question- Differentiability on an interval

So this is the question I am trying to answer... At $(-2,0)$ and $(0,2)$ we just differentiate using normal rules of calculus, yes? Here is my attempt for at $x=0$. Is this correct? For d)I think ...
0
votes
0answers
27 views

Continuous convergence [on hold]

If f_n converge pointwise to $0$ in $\mathbb{R}^d$, $\int f_n dm =1$ for every $n\in \mathbb{N}$ and $g \in L^1_m \cap C(\mathbb{R}^d,\mathbb{R})$. Then how can I prove that: \begin{equation} \int ...