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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0answers
251 views

Riemann-Stieltjes Integral computation (step function?)

Im trying to integrate this, using theorem 7.9 of apostol's book: $$\int^{10}_0 f(x)d\alpha(x) $$ $f(x) = x^2$ and $\alpha(x)= 3\chi(7,9](x)$ Where $\chi(x)$ is $0$ everywhere except $1$ in the ...
4
votes
2answers
24 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
0
votes
1answer
23 views

New variable in a convex optimization problem

Consider the convex optimization program $$ \min_{x \in X } x^\top P x + p^\top x \quad \text{ sub. to: } Ax = b $$ where $X \subset \mathbb{R}^n$ is compact, $P \succ 0$, $A \in \mathbb{R}^{m \times ...
0
votes
2answers
435 views

Restriction of a Lebesgue integral to a subset of a measurable set.

Let $f$ be a bounded measurable function on a set of finite measure $E$. For a measurable subset $A$ of $E$, show that $\int_A f = \int_E f \cdot \chi_A$ Lemma: Let $f$ be a bounded measurable ...
1
vote
1answer
243 views

Uniform convergence of integral when integrand converges pointwise

Let $g(x)$ be a continuous real valued function defined on $[-a,1+a]$ where $a>0.$ Let $f(x)$ be a continuous real valued function, $f\left( x\right) \geq 0$ for $x\in \lbrack 0,1],$ equal to ...
4
votes
4answers
122 views

Real Analysis: Showing $f: \Bbb Q \to \Bbb Q$ is continuous

The following is all working in $\mathbb{Q}$, not $\mathbb{R}$. I am working with the function $f: \mathbb{Q} \to \mathbb{Q}$ defined piece-wise by $f(x)=-1$ if $x^2<2$ $f(x)=1$ if otherwise I ...
0
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0answers
27 views

asymptotical behavior of integral

I'm interest in the asymptotical of $$\int_{-\pi}^{\pi}\exp\Big((\cos z+i\alpha\sin z-1)t\Big)dz\hspace{3mm}\text{as}\hspace{2mm}t\to\infty$$ for $-1<\alpha<1$. Numberical result suggest that ...
5
votes
1answer
312 views

Maps that preserve measure zero property

Consider a map $f: \mathbb{R}^n \to \mathbb{R}^m$ that is differentiable (usually even smooth). If $B \subset \mathbb{R}^m$ has measure zero (Lebesgue measure), then what types of maps $f$ satisfy $A ...
1
vote
1answer
29 views

Uniform and integral limit

Let $f_n(x)=n(\sin x)^n \cos x$. Show that the sequence of functions $f_n$ converges to $0$ uniformly on any interval of the form $[0,a]$ where $a<\pi /2$. Show that, for any continuous function ...
2
votes
5answers
94 views

How I could show that :$\log1=0$?

I would be like somone to show me or give me a prove for this : Why $\ln 1=0$ ? Note that $\ln$ is logarithme népérien, the natural logarithm of a number is its logarithm to the base $e$. Thanks ...
2
votes
1answer
50 views

A question on limit of a sequence

Suppose that $\varphi(n)$ is a positive monotone increasing function defined on $N$ and $\lim_{n\to \infty}\frac{\varphi(n)}{n}=0$. Let $\{n_k\}$ be a subsequence with $\lim_{k\to ...
0
votes
0answers
28 views

Seeing that a function is a trigonometric polynomial

I'm working through Chapter 4 of Rudin's Real and Complex Analysis book right now, and I've found myself rather more confused than usual. In the proof of the completeness of the trigonometric system, ...
3
votes
0answers
26 views

Convergence of measures — revisited

In this thread, I asked a question about the convergence of measures. The conjecture I posed there, which turned out to be false, was supposed to be a lemma that I wanted to use to prove a ...
2
votes
0answers
36 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
0
votes
1answer
31 views

Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?

According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if: Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with ...
2
votes
1answer
72 views

Does the derivative of a bounded smooth monotone function have a limit at infinity?

Let $f \in C^1(\mathbb{R})$ a monotonic function such that $$\lim_{x \to \infty} f(x) = m \in \mathbb{R}$$ Does this imply $\displaystyle\lim_{x \to \infty} f'(x) = 0$? If so, can the hypothesis be ...
0
votes
0answers
27 views

Relation between measurable sets

Given an outer measure $\mu^*$ with domain $X$ and a subset $B$ of $X$, one can construct a new outer measure $\nu^*(A) = \mu^*(A \cap B)$. The problem is to find the relation between the measurable ...
2
votes
1answer
65 views

Points in a general Cantor set

We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine ...
1
vote
1answer
35 views

What smooth functions are solutions of an autonomous ODE?

Let $y$ be a smooth function, say $y : \mathbb{R} \rightarrow \mathbb{R}$. When can we find a continuous map $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $y'=f(y)$ ? Obviously it's not always ...
0
votes
2answers
29 views

Find a function $f$ which is $L^p$ integrable but not $L^\infty$ integrable satisfies $\int_{supp(f)} e^{|f|} dx =1$

Find a function $f$ which is $L^p(\mathbb{R})$ integrable but not $L^\infty(\mathbb{R})$ satisfies $\int_{supp(f)} e^{|f|} dx =1$. I have no idea how to find such function belongs to $L^p\cap ...
63
votes
3answers
3k views

Is there an integral that proves $\pi > 333/106$?

The following integral, $$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which proves ...
31
votes
2answers
1k views

Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$

Some time ago when decomponsing the natural numbers, $\mathbb{N}$, in prime powes I noticed a pattern in their powers. Taking, for example, the numbers $\lbrace 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 ...
3
votes
1answer
26 views

Show $p'(x) + kp(x)$ has a zero between consecutive roots of the polynomial $p$

For any $k \in \mathbb{R}$ and any polynomial $p(x)$, show that $$p'(x) + kp(x)$$ has a zero between consecutive roots of $p$. I have tried writing $p(x) = a_n x^n + \cdots + a_0$, but this does not ...
4
votes
1answer
31 views

Proving this function is differentiable at $1$

Define $h(x) = 1$ except at $1$ where $h(1) = 0$. Also define $H(x) = \int_0^x h(t)$. Now I tried to show that $H$ is differentiable at $1$. My proof is to compute $$ \lim_{x \to 1^-} {H(1) - ...
0
votes
0answers
19 views

differential equation with random coefficient

I am confused with a problem I encountered at hand, not on how to work on it but rather understanding the problem itself: Let $A(x;\omega)$ be a random field taking values in $[a,b]$ where $a,b < ...
2
votes
1answer
43 views

Convergence of Taylor series of $\sqrt{1-x}$

Concerning $$\sqrt{1-x} = \sum_{k=0}^{\infty} \left[\prod_{j=1}^k \left(\frac{j-1-\frac{1}{2}}{j}\right)\right]x^k$$ the Taylor series about $x=0$. For $|x|< 1$ this series converges uniformly. ...
1
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0answers
19 views

On seqeunce of functions $h_n$ satisfying $\Vert\sum_{n=1}^\infty f\ast h_n\Vert_1=\sum_{n=1}^\infty\Vert f \ast h_n\Vert_1$ for all $f\in L_1(G)$

Let $(h_n)$ be a sequence of non-zero functions in $L_1(G)$ (where $G$ is a locally compact group) with the property $$ \left\Vert\sum_{n=1}^\infty f\ast h_n\right\Vert_1=\sum_{n=1}^\infty\Vert f ...
1
vote
2answers
33 views

For $f, g \in C^1$, $fg' - f'g \neq 0$ implies that the zeros interlace

Let $f, g \in C^1$, and suppose that $f(x) g'(x) - f'(x) g(x) \neq 0$ for all $x$. Show that The roots of $f$ do not have an accumulation point. The roots of $f$ and $g$ interlace, so that if ...
0
votes
1answer
26 views

is bounded partial derivative continous

Let $f:{\mathbb R}^2\rightarrow {\mathbb R}$ be defined as: $$ f(x,y) = \left\{ \begin{array}{ll} \frac{x^3}{x^2 + y^2}, & \ (x,y)\ne(0,0),\\ 0, & \ (x,y)=(0,0).\\ \end{array} \right. $$ Prove ...
1
vote
2answers
22 views

Text on convergence theorems in probability theory (various modes of convergence)

I need a text reviewing theorems and discussing with details ALL the types of convergence in probability theory such as almost sure convergence, convergence in probability, weak convergence, $L^p$ ...
7
votes
2answers
49 views

Integral limit of $\sin(x/n)f(x)$

For any $f\in L^1[0,\pi]$, evaluate $n\to \infty \int^\pi_0 n$sin$(x/n)f(x)dx$ My idea is, $n$sin$(x/n)f(x)\to xf(x)$ and it seems that it is increasing sequence. I am not able to show it is ...
0
votes
2answers
34 views

What functions satisfy the condition $f(x,y)=g(x)$?

Are there any functions $f(x,y)$ and $g(x)$ that satisfy (1) $f(x,y)=g(x)$ for all $x \in \mathbb{R}$ and $y\in \mathbb{R}$ (2) $f(x,y)$ is not constant in $y$ for each $x$ (i.e. for each $x$ there ...
9
votes
8answers
229 views

Proof that if $a_1=1$ and $a_{n+1}=1+\frac{1}{1+a_n}$

Question: Proof that if $$a_n=\left\{ \begin{array}{ll} a_1=1\\ a_{n+1}=1+\frac{1}{1+a_n} \end{array} \right.$$ then $a_n$ converge, and then find $\lim_{n \to \infty}a_n$. I found ...
0
votes
1answer
42 views

Homeomorphism are equivalence relations, so what are the equivalence classes?

Homeomorphisms are equivalence relations, so what are the equivalence classes for two Topological spaces $T_1, T_2$? Intuitively it seems like we might have the following equivalence classes - ...
0
votes
2answers
42 views

Is a continuous function $f$ from a metric space $(X, d)$ to $\mathbb{R}$ compact under some certain condition?

$(X, d)$ is a metric space with metric $d$ and there is $x_0 \in X$ and define $E_\epsilon=\{x\in X, d(x, x_0)\geq \epsilon\}$. If $f$ is continuous function from $X$ to $\mathbb{R}$ such that ...
0
votes
1answer
52 views

$f(A) \cap f(B) = f(A \cap B)$ if $f$ is a bijection?

I found this statement in a Topology proof - $$f(A) \cap f(B) = f(A \cap B)$$ if $f$ is a bijection I haven't come across this statement before. Is this some axiom of set theory?
0
votes
1answer
36 views

Intuition behind homeomorphism from $B((0, 0), 1) \to \mathbb{R^2}$

In my notes I have that the following function is a homemorphism from $B((0, 0), 1) \to \mathbb{R^2}$ $$h(x, y) \to \frac{f(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}} (x, y)$$ where $f = ...
1
vote
3answers
38 views

Show $A$ is unbounded given $\int_A f'(x) dx \leq 0$

Let $A\subset \mathbb{R}$ be a non-empty open set such that $$\int_A f'(x) dx \leq 0$$ for all $f\in\mathcal{C}_c^1(\mathbb{R})$ with $f\geq 0$. Prove that $A$ is unbounded. The hint says first show ...
0
votes
0answers
41 views

Is $\sin (\mathbb N)$ dense in $[-1,1]$ ? [duplicate]

Let $\mathbb N$ be the set of +ve integers , then is it true that $\sin (\mathbb N)$ is dense in $[-1,1]$ i.e. is it true that for every $x,y \in [-1,1]$ with $x<y$ , $\exists m \in \mathbb N$ such ...
14
votes
1answer
557 views

Integral$=-\frac{4}{3}\log^3 2-\frac{\pi^2}{3}\log 2+\frac{5}{2}\zeta(3)$

Hi I have been trying to prove this $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
0
votes
0answers
20 views

Lipschitz at a point implies Lipschitz in a neighborhood

A function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is said to be Lipschitz at a point $x \in \mathbb{R}^n$ if there is a neighborhood $N_x$ of $x$ and $L > 0$ such that $$ |f(y) - f(x)| \leq L ...
14
votes
2answers
451 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
0
votes
2answers
11 views

A cycloid that goes through the beginning and through a general point

Parametric equations of the general cycloid through the beginning $(0,0)$ are $$x(t)=\frac{2t-\sin2t}{2d}$$ $$y(t)=\frac{1-\cos 2t}{2d}$$ How can we determine $d$ such that the cycloid goes through ...
5
votes
3answers
65 views

Is the set $\{ x\in \mathbb{Q}: 2< x^2 <3\}$ closed, bounded, compact in $\mathbb{Q}$?

Is the set $\{ x\in \mathbb{Q}: 2< x^2 <3\}$ closed, bounded, compact in $\mathbb{Q}$ ? I think $\{ x\in \mathbb{Q}: 2< x^2 <3\}=\{ x\in \mathbb{Q}: 2\leq x^2 \leq 3\}$, so it is bounded ...
0
votes
1answer
20 views

Unique solution for $\int_x^1 f(t) dt = 2x$ and $|x| < \epsilon$

Let $f$ be continuous on $\mathbb{R}$ such that $$f(0) \neq -2 \quad\text{ and } \quad \int_0^1 f(t) = 0.$$ Show that there exists $\epsilon > 0$ such that the equation $$\int_x^1 f(t) dt = 2x$$ ...
1
vote
2answers
33 views

Convergence of measures

Consider the measurable space $(\mathbb R,\mathscr B_{\mathbb R})$. Suppose that $\{\mu_n\}_{n=1}^{\infty}$ is a sequence of finite Borel measures and $\mu$ is a finite Borel measure on $\mathbb R$. ...
0
votes
0answers
29 views

A variant of the Riemann Integral

This question is related to this one. Let $S_k$, $k\in\mathbb{N}$ be a sequence of finite sets where $S_k\subset S_{k+1}\subset[0,1]$. Fixed $s$ in $S_k$, let $s'$ denote the predecessor of $s$ in ...
1
vote
0answers
46 views

Integral of an integrable function is continuous

Let $f: [a,b]\to \mathbb R$ be Reimann integrable. Define $G(x) = \int_a^x f$. Then $G$ is continuous. I tried to prove this. Can you please tell me if my proof is correct? Let $\varepsilon > 0$. ...
1
vote
1answer
41 views

Kantorovich Theorem example

I need to write in C a program that finds roots of a 6th order polynomial. I was thinking of using Kantorovich Theorem convergence of Newton's method to find when can I use Newton-Rhapson method. I'm ...
1
vote
2answers
47 views

f(x,y) jointly differentiable

What is the definition of "jointly continuously differentiable function"? I.e. when $f:\mathbb{R}^2\rightarrow \mathbb{R}:(x,y)\mapsto z$ is jointly continuously differentiable in $x,y$ ? Is it ...