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, integration through the fundamental theorem of calculus.

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4
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1answer
224 views

Constructive proof of a problem from the book Analysis by Terence Tao

Here is a problem from the book Analysis by Terence Tao: (Vol 1, Exercise 5.5.2) Let $E$ be a non-empty subset of $R$, let $n \geq 1$ be an integer, and let $L < K$ be integers. Suppose that ...
2
votes
1answer
63 views

Intuition behind closed subsets of a metric space?

Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space. Consider a metric space $$(X,d)$$ Then consider a subset of this space$$F$$ What the book ...
2
votes
3answers
119 views

$f(x)=|\cos x|+|\sin(2-x)|$ at which of the following point $f$ is not differentiable?

$f(x)=|\cos x|+|\sin(2-x)|$ at which of the following point $f$ is not differentiable? 1.$\{(2n+1){\pi\over2}\}$ 2.$\{n\pi\}$ 3.$\{{n\pi\over 2}\}$ 4.$\{n\pi+2\}$ in all cases $n\in\mathbb{Z}$ ...
5
votes
3answers
190 views

$ \int^{\infty}_0 |\frac{1}{(1+x)\sqrt x}|^p ~ \mathrm dx < \infty \implies p=?$

If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that $$ \int^{\infty}_0 |f(x)|^p dx < \infty $$ The integral is with respect to lebesgue measure. Any solution or hints would ...
2
votes
1answer
141 views

Uniform convergence and uniform boundedness

I try to understand a demonstration from a book, but I have a problem with a line. We have the series $$u(x,t) = \sum_{k=0}^\infty \frac{g^{(k)}(t)}{(2k)!}x^{2k} \qquad (*)$$ where $$g(t) = \left\{ ...
1
vote
2answers
177 views

Riemann integral and Lebesgue integral

$f:R\rightarrow [0,\infty)$ is a Lebesgue-integrable function. Show that $$ \int_R f \ d m=\int_0^\infty m(\{f\geq t\})\ dt $$ where $m$ is Lebesgue measure. I know the question may be a little dump. ...
2
votes
1answer
45 views

Does $f(x)=x^{2}\sin\left(\frac{1}{x^2}\right)$ satisfy the relation $f(x)+f(y)−2f\left(\frac{x+y}{2}\right)=O\left(\left|x−y\right|^2\right)$?

Does $f(x)=x^{2}\sin\left(\frac{1}{x^2}\right)$, $x\in(0,1)$ satisfy the relation $f(x)+f(y)−2f\left(\frac{x+y}{2}\right)=O\left(\left|x−y\right|^2\right)$?
2
votes
1answer
36 views

Linear independence of $(x\sin x)^{\frac{n-1}{2}}$ and $(x\sin x)^{\frac{n+1}{2}}$

Could you tell me why $(x\sin x)^{\frac{n-1}{2}}$ and $(x\sin x)^{\frac{n+1}{2}}$ are lineraly independent? I've tried $\alpha(x \sin x)^{\frac{n-1}{2}} + \beta (x\sin x)^{\frac{n+1}{2}} =0$ ...
1
vote
1answer
149 views

Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k

In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
3
votes
1answer
321 views

Discontinuity in function; order of integration matters

I'm struggling a bit with Chapter 10 of Rudin's "Principles of Mathematical Analysis," and I was hoping to get some help here. I'll post the problem and my current progress. Exercise 2: For $i = 1, ...
3
votes
3answers
187 views

Show that f is uniformly continuous

I am having difficulty determining what value I should assign to $\delta$ for the following problem. How do I determine what it should be? Define $f:[3.4,5] \rightarrow \mathbb{R}$ by ...
1
vote
1answer
46 views

Does $f(x)=x^{2}\sin(\frac{1}{x^{2}})$ satisfy the relation $f(x)+f(y)-2f(\frac{x+y}{2})=O(|x-y|^{2})$?

Does $f(x)=x^{2}\sin(\frac{1}{x^{2}})$ satisfy the relation $f(x)+f(y)-2f(\frac{x+y}{2})=O(|x-y|^{2})$? I can't check it. Who will hint it? Please.
2
votes
0answers
54 views

Symmetry between differentiation and integration [duplicate]

I want to make clear, that I am interested in the question: Why does integration need a bigger spectrum of functions than differentiation and not why integration is harder!!! as experience told me, ...
0
votes
1answer
82 views

power series quotient of polynomial functions

I have given $g(x)=\sum_{k=1}^\infty k^2x^k$. Why can you now write $g:(-1,1)\rightarrow\mathbb R$ as a quotient of two polynomial functions? I just know the radius of convergence is ...
1
vote
1answer
63 views

Lipschitz condition normed vector space

Am I right that $g: C^{1}[0,2],||*||_{C^1[0,2]} \rightarrow \mathbb{R}$ with $g(f)=f'(1)$ satisfies a Lipschitz condition? Cause $|f'(1)-h'(1)|=|g(f)-g(h)|\le \text{max} |f-h|+ \text{max} |f'-h'|$, ...
1
vote
1answer
76 views

Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.

I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
2
votes
2answers
392 views

Continuous functions on $[0,1]$ is dense in $L^p[0,1]$ for $1\leq p< \infty$

I tried to show that the continuous functions on $[0,1]$ are dense in $L^p[0,1]$ for $ 1 \leq p< \infty $ by using Lusin's theorem. I proceeded as follows.. By using Lusin's theorem, for any $f ...
4
votes
2answers
730 views

Proof of Egoroff's Theorem

Let $\{f_n \}$ be a sequence of measurable functions, $f_n \to f$ $\mu$-a.e. on a measurable set $E$, $\mu(E) < \infty$. Let $\epsilon>0$ be given. Then $\forall \space n \in \mathbb{N} \space ...
1
vote
1answer
94 views

Continuous function on a closed set

Let $f: F \to \mathbb R$ be defined in a closed set $F \subset \mathbb R$. Show that $f$ is continuous if and only if for all $c \in \mathbb R$, the sets $E[f \le c]=\{x \in F; f(x) \le c\}$ and $E[f ...
0
votes
1answer
89 views

Calculation of a multivariable integral

I read a paper which contains a tedious calculation and end up with the following integral: $$ \int_{\mathbb{R^2}}\int_{\mathbb{R^2}}\frac{1}{1+\|x\|^2}\frac{1}{1+\|x+y\|^2}\frac{1}{1+\|y\|^2} \, ...
1
vote
2answers
68 views

Question on Contractions

Let $S = \{x \in \mathbb{R}^n ; \|x\| \le 1 \}$ and $f: S \to S$ be a contraction. Determine one can have $f(S) = S$. I really need some help with this question. In advance I wanted to give all ...
3
votes
2answers
96 views

Searching for unbounded, non-negative function $f(x)$ with roots $x_{n}\rightarrow \infty$ as $n \rightarrow \infty$

If a function $y = f(x)$ is unbounded and non-negative for all real $x$, then is it possible that it can have roots $x_n$ such that $x_{n}\rightarrow \infty$ as $n \rightarrow \infty$.
4
votes
1answer
58 views

Any arbitrary closed smooth curve bounds a orientable surface?

I've got a question that, given an arbitrary closed smooth curve $C:[0,1]\rightarrow\mathbb{R}^3$, can you always find a orientable surface $\Omega$ which satisfy $\partial\Omega=C[0,1]$ ? I have no ...
1
vote
0answers
70 views

Antiderivative of an absolute function

$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$ $$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) \lim_{x \to ...
2
votes
1answer
65 views

Finding the maximum and minimum of $f$ on a set $Q$

I stumbled across a proposed task that I'm unable to solve. We have a function $f:\mathbb{R^2}\to\mathbb{R}$ defined as: $$ f(x,y):=x^2+xy+y^2+x+y+1 $$ The task is to "find the maximum and minimum" ...
6
votes
4answers
218 views

Open Cover / Real Analysis [duplicate]

I have the next question: Let $K \subset $ $R^1$ consist of $0$ and the numbers 1/$n$, for $n=1,2,3,\ldots$ Prove that $K$ is compact directly from the definition (without using Heine-Borel). I'm ...
1
vote
0answers
28 views

Example on Correspondences

Give an example of correspondences F: X $\rightarrow$ Y, G: Y $\rightarrow \mathbb{R}^s$ such that F and G are closed, but (G o F) is not, if any, where $ \varnothing \neq X \subset\mathbb{R}^m, ...
2
votes
2answers
102 views

which of the followings are true for bijective functions

which of the followings are true:- 1. There is a continuous bijection from $\mathbb{R}^2\to \mathbb{R}$. 2. There is a bijection between $\Bbb{Q}$ and $\mathbb{Q}\times \mathbb{Q}$. Can somebody ...
6
votes
2answers
490 views

Diffeomorphism from Inverse function theorem

I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
1
vote
1answer
403 views

How to know a function is integrable or not?

Let say $$ h(x) = \begin{cases}x^2,& x \in \mathbb {Q}\\-x^2,& x \notin \mathbb{Q}\end{cases} $$ Is there a difference between riemann integrable and integrable? And can I just ...
3
votes
2answers
99 views

Rudin Theorem 1.17

1.17 THEOREM Let $f: X \to [0, \infty)$ be measurable. There exist simple measurable functions $s_n$ on $X$ such that (a) $0 \le s_1 \le s_2 \le \dots \le f.$ (b) $s_n(x) \to f(x)$ as $n \to \infty ...
1
vote
1answer
441 views

Is every convergent sequence Cauchy?

Wikipedia: "Every convergent sequence (with limit s, say) is a Cauchy sequence, since, given any real number ε > 0, beyond some fixed point, every term of sequence is within distance ε/2 of s, so any ...
2
votes
0answers
178 views

Helly's selection theorem

Can someone guide me to a reference (preferably open access online) stating and proving Helly's selection theorem for sequences monotone uniformly bounded functions on $[0,1]$. Something that can ...
1
vote
2answers
48 views

How can i show this inequality?

Let $n>1$ and $a_1,...,a_n \in \mathbb{R}^+$ be such that $\sum a_i=1$. For evey $i$, define $b_i=\sum_{j=1,j\neq i}a_j$. Show that $\sum_{k=1}^n \dfrac{a_k}{1+b_k}\ge \dfrac{n}{2n-1}$ Thanks a ...
0
votes
0answers
41 views

If a function $f:J\to\mathbb{R}$ satisfies the Zygmund condition, is it $C^1$?

A function $f\colon J\rightarrow \mathbb{R}$ on an open interval $J$ satisfies Zygmund condition if, for all $x,y\in J$, $$f(x)+f(y)-2f\left(\frac{x+y}{2}\right)=o(|x-y|).$$ It is clear, if $f\in ...
1
vote
1answer
89 views

Is this set relatively open or closed?

Take the set $\{(x,y) : (x-2)^2 + y^2 < 2\}$ . Is this set relatively open or relatively closed in the subspace $B(2,0)$ radius $\sqrt2$. (The open ball centered at $(2,0)$ with radius $\sqrt2$) ...
2
votes
1answer
138 views

limsup and liminf and the product of sequences

I'm trying to show that if $ \limsup s_n = +\infty$ and $\liminf t_n > 0$, then $\lim\sup s_n t_n = +\infty$. Could someone check my proof/give feedback? Since $\lim\inf t_n > 0$, we know that ...
1
vote
1answer
234 views

Composition of Lebesgue measurable function $f$, with a continuous function $g$ having a certain property, is Lebesgue measurable

Suppose that $f$ is Lebesgue measurable and $g$ is real valued, continuous, and has the property that for any null set $N$, $g^{-1} (N)$ is measurable. Then $f \circ g$ is also Lebesgue measurable. ...
4
votes
4answers
753 views

How is the boundary of the clopen set [0,1) empty?

I don't get why the boundary of a clopen set is empty. If you take A = [0,1) in R, then isn't the closure of this the smallest closed super set that contains A..which is [0,1]. Isn't the interior, ...
-1
votes
2answers
93 views

If $f: \mathbb{C}\to\mathbb{C}$ is bounded, then is it a constant? [closed]

If a function $f: \mathbb{C}\to\mathbb{C}$ is bounded, then it is a constant. Is it true or false?
2
votes
1answer
153 views

Weierstrass $M$-test problem, $f_n(x)=(nx^2)/(n^3+x^3)$

Use the Weierstrass M-test to show $$f(x)=\sum_{n=1}^\infty \frac{nx^2}{n^3+x^3}$$ converges uniformly on any finite interval $[-R,R]$. This was an exam question I had. My attempt was to find an ...
0
votes
3answers
181 views

I'm just curious, what exactly is $\mathbb{R}\setminus\mathbb{Q}$? [duplicate]

What exactly is $\mathbb{R}\setminus\mathbb{Q}$? How many different kinds of things live in this place? For $n>1$ how does $$ q_1x_1+\cdots+q_nx_n=p $$ have a solution for $q_i,p\in \mathbb{Q}$ ...
4
votes
1answer
108 views

What is the formulation of the Least Upper Bound propierty in First Order Logic?

I've been readining about the completeness Godel's theorems. Accordingly, the axioms of $R$ in first order logic make up one of these sets that is complete and consistent. But I've always seen the ...
1
vote
1answer
46 views

Recurrence inequality for Dirichlet's eta function.

I'm studying the following function: $\theta(p)=\eta(p)\eta(p-2)-\frac{p-1}{p}\eta^2(p-1)$, where $\eta$ - Dirichlet's eta function. If we build a plot for $p\in [1,150]$, it's easy to see that it's ...
0
votes
1answer
126 views

Riemann integral show $f(x)=g(x)$ for at least 1 $x$ in [a,b]

Let $f$ and $g$ be continuous functions on $[a,b]$ such that $\int_a^b f = \int_a^b g$. Show that there exists $x\in [a,b]$ such that $f(x) = g(x) $. I want to assume not and then show that the ...
5
votes
0answers
136 views

Is there a subsequence of $a_n = n \sin(n)$ which tends to $0$?

I know there is such a subsequence for $b_n = \sin(n)$. What about $a_n = n\sin(n)$?
1
vote
1answer
139 views

Monotonic integral proof

Let $f$ be a continuous function on $[a,b]$ such that $f(x) \geq 0 $ for every $x\in [a,b]$. Suppose $\int_a^b f = 0$ and show that $f (x) = 0$ for every $x\in [a,b]$. obv this is monotonic ( ...
11
votes
2answers
1k views

Does uncountable summation, with a finite sum, ever occur in mathematics?

Obviously, “most” of the terms must cancel out with opposite algebraic sign. You can contrive examples such as the sum of the members of R being 0, but does an uncountable sum, with a finite sum, ...
0
votes
1answer
107 views

Changing order of derivatives

I would like to rewrite the following expression $$\frac{d^i}{dx^i}\left\{f(x)\left[\frac{d^jf(x)}{dx^j}\right]\left[\frac{d^kf(x)}{dx^k}\right]\right\}$$ into the form $$D f(x)^3,$$ with $D$ ...
0
votes
1answer
162 views

Liouville's formula

I have some questions concerning a proof of Liouville's formula: $$W'(t)=\text{tr}(A) W(t)$$ where $W$ is the Wronskian of the homogenous ODE. If the vectors in the columns of the fundamental matrix ...