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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10
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2answers
62 views

$f$ be a smooth function on real line , $f(0)=0$ , $f(x)>0, \forall x \ne 0$ and any $f^{(n)}(0)=0$ ; is $\sqrt f$ smooth?

Let $f: \mathbb R \to \mathbb R$ be an infinitely differentiable function such that $f(0)=0$ , $f(x)>0 , \forall x \ne 0$ and $f^{(n)}(0)=0$ ( the $n$-th derivative ) $, \forall n \in \mathbb N$ ...
4
votes
0answers
18 views

What is the regularity of the greatest eigenvalue of the Hessian matrix?

Let $f:\mathbb{R}^n\to\mathbb{R}$ be twice continuously differentiable, i.e. $f\in\mathcal{C}^2\left(\mathbb{R}^n\right)$. Define $\lambda_f:\mathbb{R}^n\to\mathbb{R}$ as the function which associates ...
0
votes
1answer
23 views

Diffuse-like decomposition of the segment $[0,1]$

Consider the segment $[0,1]\subset\mathbb{R}$ and the standard Lebesgue measure $\mu$ on $\mathbb{R}$. I wonder if we can find such decomposition $A\sqcup B=[0,1]$, that for any $x\in[0,1]$ we have $\...
-2
votes
0answers
17 views

Each convergent sequence of functions converges to a same limit, the whole sequence of function converges to that limit

Say I have a sequence of function $f_n$, which is real and bounded. I have proved for each convergent sequence, $f_{nk}$ converges to L, does $f_n$ converges and converges to L? Can any one provide ...
-1
votes
0answers
13 views

Real Analysis, Folland Theorem 2.26 Integration of Complex Functions

Background information: Theorem 2.10 - Let $(X,M)$ be a measurable space. a.) If $f:X\rightarrow [0,\infty]$ is measurable, there is a sequence $\{\phi_n\}$ of simple functions such that $0 \...
1
vote
2answers
31 views

Existence of Continuous bijective function

I was looking for examples of bijective continuous functions infollowing cases Does there exist Bijective continuous function $(0,1) \rightarrow \mathbb{R}$? Yes, $f(x)=\frac{2x-1}{x-x^2}$ Does ...
1
vote
0answers
20 views

Real Analysis, Folland Theorem 2.25 Integration of Complex Functions

Theorem 2.25 - Suppose that $\{f_j\}$ is a sequence in $L^1$ such that $\sum_{1}^{\infty}\int |f_j| < \infty$. Then $\sum_{1}^{\infty}f_j$ converges a.e. to a function in $L^1$, and $$\int \sum_{1}^...
3
votes
1answer
22 views

Real Analysis, Folland Proposition 2.22 Integration of Complex Functions

Proposition 2.22 - If $f\in L^1$, then $|\int f|\leq \int |f|$ Attempted proof - If $f$ if a real-valued function then $$\left|\int f\right| = \left|\int f^+ - f^-\right|\leq \int f^+ + \int f^- = \...
2
votes
2answers
62 views

Find $\lim\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}$

The question arise in connection with this problem Prove that $$\lim_{n\rightarrow \infty}\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\...
1
vote
0answers
26 views

Real Analysis, Folland The Dominated Convergence Theorem

Background Information: Proposition 2.16 - If $f\in L^+$, then $\int f = 0$ iff $f = 0$ a.e. Question: 2.24 The Dominated Convergence Theorem - Let $\{f_n\}$ be a sequence in $L^1$ such that ...
-3
votes
2answers
71 views

$\lim\limits_{x\mapsto \infty}\int_0^x| f (x, t)|dt = 0$ implies $\lim\limits_{x\mapsto \infty} f (x, t) = 0$

$\lim\limits_{x\mapsto \infty}\int_0^x |f (x, t)|dt = 0$ implies $\lim\limits_{x\mapsto \infty} f (x, t) = 0$ for all $ t$. $\quad$$ f (x, t) $ is jointly continuous in the variables $ x, t $ $ x \...
0
votes
0answers
10 views

Real Analysis, Folland Proposition 2.21 Integration of Complex Functions

Proposition 2.21 - The set of integrable real-valued functions on $X$ is a real vector space, and the integral is a linear functional on it. Attempted proof - Note that we can derive the axioms of a ...
3
votes
0answers
69 views

how to solve this functional equation $\int_{x}^{x+1}{f(y)dy}=f(x)$ for all x

Suppose $f(x)$ is continuous and bounded on $\mathbb{R}$ and $\int_{x}^{x+1}{f(y)dy}=f(x)$ for all x. Prove $f(x)$ is constant on $\mathbb{R}$.
4
votes
1answer
53 views

Measurable function and the Mean Value Theorem

Let $\,f:[a,b]\to \mathbb{R}\,$ be continuous on $[a,b]$ and derivable on $(a,b)$. By the mean value property, for all $\,x\in (a,b)\,$ there exists $\,\xi_x\in (a,x)\,$ such that $\,f(x)-f(a)=f'\left(...
2
votes
1answer
143 views

Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite subcover.

I really need help with this question: Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite subcover.
1
vote
1answer
154 views
+100

Theorem regarding Change of Variables in finite dimesnion

My question is based on Change of Variables in Multiple Integrals II Peter D. Lax > It is not necessary to read the paper before answering this question.The author tried to prove change of variables ...
3
votes
2answers
70 views

Tricky detail in extreme value theorem proof

I am reading Pugh's Real Mathematical Analysis, and in chapter 1, section 6, ``The Skeleton of Calculus,'' Pugh supplies a proof of the Extreme Value Theorem. I am having trouble understanding one ...
6
votes
2answers
79 views

How to prove this statement? (Real analysis)

This might be the basic question in real analysis. A function $f$ is $ C^2 $ function on the closed interval$ [0,1]$ Also the function $ f $ is satisfying $ f(0) = f(1) =0 $ Plus, $\vert f''(x) \...
1
vote
1answer
31 views

what is the definition of the space $C([0,T];H^s)$?

What is the definition of the space $C([0,T];H^s)$? Here, we are considering the solutions of a PDE, and $H^s$ is the Sobolev space. My book says we are assuming that a solution lies in this space, ...
0
votes
0answers
10 views

Definition of the space $H^s(\mathbb{R}^n)$

The following is the definition of the space $H^s(\mathbb{R}^n)$ in Hunter's Applied Analysis: Here a regular distribution is a tempered distribution $T_f$ such that it is given by $$ T_f(\varphi)=\...
2
votes
0answers
28 views

Creating a tight frame of $\mathbb{R}^{n}$ when already knowing some of its vectors.

I'm wondering whether or not it's possible to start with a matrix $S\in\mathbb{R}^{m\times mn}$, $m<n$, and add rows to it so that the columns of the resulting matrix form an orthogonal system of ...
0
votes
1answer
31 views

Darboux integral epsilon-delta proof in piecewise continuous function

If $f$ is a piecewise continuous function in $[a,b]$, then It is integrable in $[a,b]$. Furthermore, given $\epsilon>0$, there is $\delta>0$ such that for every $P$ partition: $$||P|| < \...
0
votes
0answers
8 views

Showing that $\log (\log 1+\frac{1}{|x|})$ belongs to $W^{1,p}(\Omega)$ for $p \geq 2$.

I want to show that the function $f$ belongs to $W^{1,p}(\mathbb{R}^n)$ for $p \geq 2$, where $f$ is defined as $$ f(x)=\log\left(\log \left(1+\frac{1}{|x|}\right) \right)$$ Note: This is an example ...
4
votes
1answer
3k views

Bounded sequence and every convergent subsequence converges to L

Let $\{x_n\}$ be a bounded sequence such that every convergent subsequence converges to $L$. Prove that $$\lim_{n\to\infty}x_n = L.$$ The following is my proof. Please let me know what you think. ...
-1
votes
0answers
22 views

For a compact metric space $X$, does the subset of surjective continuous maps $X\to X$ have non-empty interior? [on hold]

Let $(X,d)$ be a compact metric space and denote with $S(X)$ the set of all continuous maps $f:X\rightarrow X$. If $d_U(\varphi, \psi):=\sup_{x\in X}d(\varphi(x), \psi(x))$ for $\varphi, \psi \in S(...
0
votes
0answers
36 views

Probabilistic Modeling Parameters Request

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let us restrict our attention to the vectors $\overrightarrow{x} = (x_{1},x_{2},\ldots,...
0
votes
0answers
11 views

Let $\phi:(0,1)\to \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ given by $\phi(t)=A(tx)$, then $\phi'(t). h=(A'(tx). x). h$ or $(A'(tx). h). x$?

Let $U$ be an open ball centered in $0$ in $\mathbb{R}^m$. Given $\phi:(0,1)\to \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ be defined by $\phi(t)=A(tx),$ where $A:U\to \mathbb{R}^n$ and $x\in U$, which ...
0
votes
3answers
110 views

Value of the integral $\int_{a}^{a}f(x)dx?$

I am confused about the value of the integral $\int_{a}^{a}f(x)dx,$ is it $0$ or not always? In Riemann integration case (i.e. $f$ is bounded ) its zero by definition. What about other cases for ...
1
vote
1answer
47 views

Is this a proof for Heine-Borel Theorem in $R$ and if it is why doesn't this work for open sets?

$[a,b]$ has an open cover then every point of this interval is covered in an open set. Since a is in an open set some neighborhood of a is a subset of that set. Say $[a,a+r_1) \subset O_i $ similarly $...
0
votes
2answers
28 views

Proving a bounded monotone function has finite one-sided limits

all! I got stuck on this question today, although it seemed straight forward when I started. Here is the proposed problem: Let a and b be extended real numbers with a $\lt$ b. Prove that if f is a ...
0
votes
0answers
27 views

Weakening the hypothesis for integration by parts

Assume $h$ and $k$ have continuous derivatives on $[a,b]$ and then $$\int_a^b h'k=h(b)k(b)-h(a)k(a)-\int_a^bhk'$$ Explain how the fact that if $f$ and $g$ are Riemann-integrable in $[a,b]$ ...
0
votes
0answers
10 views

bounded predictable functions, generated by bounded continuous functions

Assume we have a probability space $\{\Omega,\mathcal{A},P\}$, we also look at functions (stochastic processes) $X(t,\omega): [0,\infty)\times \Omega$. We define $\mathcal{G}$ as the $\sigma$-algebra ...
15
votes
4answers
417 views
+100

How do you prove that $\{ Ax \mid x \geq 0 \}$ is closed?

Let $A$ be a real $m \times n$ matrix. How do you prove that $\{ Ax \mid x \geq 0, x \in \mathbb R^n \}$ is closed (as in, contains all its limit points)? The inequality $x \geq 0$ is ...
3
votes
1answer
39 views

Intersection of Compact sets Contained in Open Set

Just wanted to see if my proof of the following is valid: Let $\{K_i\}_{i=1}^{\infty}$ be compact sets (in some metric space), and let $V$ be an open set such that $$ \bigcap_{i=1}^{\infty} K_i \...
0
votes
0answers
15 views

Introductory Analysis: Abbott or Ross

I am attempting to self study analysis. I've heard mixed reviews on intro analysis books and it seems Abbott and Ross are proper for initial exposure. Does anyone have preference or recommendations on ...
0
votes
0answers
20 views

Riemann integrability of trigonometric functions clarifying

Consider a function which is periodic with period $2\pi$, and is Riemann integrable on the closed interval $[-2\pi,2\pi]$. Now, can we say that the function is Riemann integrable on every bounded ...
1
vote
1answer
72 views

Integration problem $\int_0^\frac{\pi}{2} \sin^n 2x\cos^n3x \, dx$

This is the problem I'm trying to solve: Find $\int_0^\frac{\pi}{2} \sin^n 2x \cos^n 3x \, dx$. For $n=0$ we get $\frac{\pi}{2}$, for $n=1$ we have $$\int_0^\frac{\pi}{2} \sin2x\cos3x \, dx = \...
1
vote
1answer
26 views

Continuity of the improper integral $\int_{0}^{x}y^{-1/2}dy.$

The improper integral $\int_{0}^{x}y^{-1/2}dy$ is $1.$ Continuous on $[0,\infty).$ $2.$ Continuous on $(0,\infty).$ $3.$ Continuous only on $[1/2,\infty)$ It is cleat that option $3$rd is wrong ...
0
votes
0answers
25 views

Bernoulli Polynomials from Apostol's calculus book

Question 35 from book of calculus, volume 1 Apostol, in chapter "The relation between integration and differentiation". Define Bernoulli polynomials as: $P_0(x)=1$, $P'_n(x)=nP_{n-1}(x)$, $\int_0^...
-1
votes
1answer
26 views

Limits superior: $\limsup_{n\rightarrow\infty}x_k=\sup\{\lim_{k\rightarrow\infty}x_{n_k}\}$?

Let $x\in \ell^\infty$. By definition we have $\limsup_{n\rightarrow\infty}x_n=\lim_{n\to\infty}\sup_{m\geq n} x_m$. Can you also write $\limsup_{n\rightarrow\infty}x_k=\sup\{\lim_{k\rightarrow\infty}...
1
vote
1answer
68 views

Using Fatou's Lemma to Prove Monotone Convergence Theorem

Monotone Convergence Theorem- If $\{f_n\}$ is a sequence in $L^{+}$ such that $f_j\leq f_{j+1}$ for all $j$, and $f = \lim_{n\rightarrow \infty}f_n(=\sup_{n}f_n)$, then $\int f = \lim_{n\rightarrow\...
-4
votes
0answers
15 views

Riemann integrability of trigonometric functions [on hold]

Are Riemann integrable functions on a specific closed interval integrable on every bounded interval
0
votes
2answers
53 views

Orthogonal of an Hilbert subspace and density

If $V$ is a subspace of an Hilbert space $H$, I know that the orthogonal of $V$, $V$$^o$, is ($V$closed)$^o$, even if $V$ is not closed. Does this mean that $V$ is always dense in $V$$^o$? Thanks!...
0
votes
1answer
20 views

Showing a 2D continuous function can be approximated by a finite sum of “simpler” continuous functions

Show that given a real-valued continuous function $f$ on $[0,1] \times [0,1]$ and an $\epsilon > 0,$ there exist real-valued continuous functions $g_1, \dots, g_n$ and $h_1 , \dots, h_n$ on $[0,...
0
votes
1answer
33 views

What vectors can be generated by permuting and halving?

$x$ is a vector in the unit simplex in $\mathbb{R}^n$, i.e: $$x = (x_1,\dots,x_n)\,\,\,\,\,\,\,\,;\,\,\,\,\forall i: x_i\geq 0\,\,\,\,;\,\,\,\,\,\,\,\,\sum_{i=1}^n x_i = 1$$ Initially, $x=(0,0,\dots,0,...
2
votes
1answer
30 views

Upper bound for $\sum_{i=1}^{N}{x_i^{\beta_i}}$

$\{x_i\}_{i=1}^{N}$ a sequence of positive real numbers and $\{\beta_i\}_{i=1}^{N} $ are real numbers such that $\underset{1 \leq i \leq N}\min{\beta_i}$ > 1. Is it possible to find an upper bound of ...
2
votes
0answers
18 views

$\sum_{n=1}^{\infty}a_nf_n(x)$ diverges on a subset of positive measure of $[0,1]$

Let $m$ denote the Lebesgue measure on $[0,1]$. Let $f_n$ be a sequence of measurable functions defined on $[0,1]$ such that $0\le f_n\le M$ for each $n$. Moreover, assume that $\int_0^1f_n(x)dm(x)=1$ ...
-1
votes
2answers
40 views

Showing a function is integrable [on hold]

Let $\xi, \zeta\in\mathbb{R}^m$. How might one try to show that $f:\mathbb{R}^m\times\mathbb{R}^m\rightarrow \mathbb{R}$, defined by $\displaystyle\frac{1}{(1+\left|\xi - \zeta\right|)^{k}}$ is or is ...