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.

learn more… | top users | synonyms

1
vote
0answers
5 views

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ ...
0
votes
0answers
10 views

Monotonic Functions and Uniform Convergence

The following is a proof from "Heavy-Tail Phenomena" by Resnick (2007). I have some questions about the proof. (2.3) seems to be an identity. The left side the global sup over $[a, b]$ and hence ...
0
votes
1answer
27 views

Compute $\lim\limits_{x \to 1}[f(x)]$ and $\lim\limits_{x \to -1}[f(x)]$ for $f(x)=\frac{1}{1-x}-\frac{1}{1+x}$

Is it possible to rewrite expression $\frac{1}{1-x}-\frac{1}{1+x}$ in order to be able to find its values near $x=1$ and $x=-1$ more precisely? This is a question in a numerical methods course. Is the ...
0
votes
0answers
17 views

Terminology: 'inverse' of non-strictly monotonic function?

Suppose I have a nonincreasing function, $x \mapsto f(x) = y$. I want to call the function $$y \mapsto g(y) \triangleq \sup\{x:f(x) \geq y\} $$ the `inverse' of $\ f$ for brevity, despite that $\ f$ ...
1
vote
1answer
12 views

Obtaining Picard Iteratives in a coupled system

Problem: Obtain the first 5 Picard´s Iteratives of the Cauchy problem: $(dx/dt) = y $ with $x(t=0)=0 $ and $(dy/dt)=-\sin(x) $ with $y(t=0)=1 $ The function $x\mapsto \sin(x)$ should be ...
0
votes
2answers
30 views

Continuous bijection from $[0,1]$ to $[0,0.5)\cup (0.5,1]$

Can we have a continuous bijection from $[0,1]$ to $[0,0.5)\cup (0.5,1]$?
0
votes
0answers
35 views

Dense subset of the plane

Consider a sequence $x_n\in \mathbb{R}^2$ with $\mathbb{Q}\times\mathbb{Q}\subset \{x_n\}$. Let $r_n\in \mathbb{R}$ a sequence of real numbers with $r_n>0$, $\sum_n r_n=1$ and $U=\bigcup_n ...
1
vote
1answer
23 views

Lower bound on $F$ under the assumption $\theta F(s)\le sF'(s)$

Let $F(s)=\int_0^{s}f(t)\,dt$. We suppose that there exists $\theta>2$ such that $\theta F(s)\le f(s)s$ for all $s\in \mathbb{R}$ and that $F(s)>0$ for all $s\in(0,+\infty)$. How to prove that ...
4
votes
2answers
33 views

Differentiability for the uniform limit of a uniformly bounded sequence of functions

Let a sequence $\{f_n\}\subset C^1(\mathbb{R})$ and $f\in C(\mathbb R)$ such that $f_n \to f$ uniformly and $f_n, f'_n$ are uniformly bounded. Question : is $f \in C^1(\mathbb R)$ ? It would be ...
1
vote
2answers
82 views

How to solve this and what is this number called? [duplicate]

What is the real number called to which the sequence $$\gamma_n =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n} - \log _e n$$ converges and what is the radius of convergence?
0
votes
1answer
18 views

$|p- \dfrac xn|>|q- \dfrac xn|$ $\implies$ $p^x(1-p)^{n-x}<q^x(1-q)^{n-x}$?

If $p,q \in (0,1)$ , and $ n \in \mathbb N$ be given and $x$ be given integer between $0$ and $n$ such that $|p- \dfrac xn|>|q- \dfrac xn|$ , then is it true that ...
1
vote
1answer
35 views

Proving a different version of MVT

$f(x)$ is differentiable in $[a,b]$, $f'(a) = f'(b) = 0$, Prove $\exists c \in (a,b)$ such that $f(c)-f(a) = (c-a)f'(c)$ What i have tried: $f(x)$ is differentiable in $[a,b]$ imply $f(x)$ is ...
0
votes
0answers
51 views

Equality case in Hölder's inequality

How can I show that $$\left(\int{p(x)^{1-\sigma}\mathrm dx}\right)^{\frac{1}{1-\sigma}}\cdot \left(\int y(x)^\frac{\sigma-1}{\sigma}\mathrm dx\right)^{\frac{\sigma}{\sigma-1}}=\int p(x) ...
3
votes
3answers
33 views

Characterization of sets of differentiability

If $f : \mathbb{R} \to \mathbb{R}$, define $C(f) = \{ x : f \text{ is continuous at } x \}$ and $D(f) = \{ x : f \text{ is differentiable at } x \}$. I have seen it proved that: $C(f)$ is a ...
0
votes
0answers
19 views

Limit of a sequence of subunitary numbers

I was wondering if the following problem has an answer: Given the sequence: $a_n = \sum_{i = 0} ^ n \frac{i}{n} \times x_i$ where $\sum_{i = 0} ^ n x_i = 0$ and $x_i \in (-1, 1), \forall i$ are ...
3
votes
1answer
33 views

If a series has the same sum under any rearrangement, then is it absolutely convergent?

Let $(V,\| \cdot \|)$ be a Banach space. Let $\{a_n\}$ be a sequence in $V$ such that $\sum a_n$ converges. Assume that for every bijection $f:\mathbb{N}\rightarrow \mathbb{N}, \sum a_n = \sum ...
2
votes
0answers
23 views

Characterisation of absolutely continuous measure on the real line

Let $\lambda, \nu$ be two Radon measures on $\Bbb R$ such that $\lambda(\Bbb R)< \infty$. Show that the following are equivalent: $\lambda \ll \nu$; $\forall \epsilon>0$ there exists ...
0
votes
2answers
26 views

For a cubic equation, prove that two critical points of the same sign imply one root

For a cubic equation of form $x^3 + p x + q = 0$, where $p < 0$, if the two critical points are of the same sign, it will have only one real root. This is easy to see from graph, but can you help ...
3
votes
2answers
57 views

Showing a set is nowhere dense in $C([0,1])$

Let $E_n$ be the set of all $f \in C([0,1])$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} |f(x)-f(x_0)| \leq n|x-x_0| \end{align*} for all $x \in [0,1]$. Why is ...
0
votes
1answer
17 views

Approximate non-Lipschitz (but continuous) functions by Lipschitz functions

Is there any algorithm to approximate non-Lipschitz (but continuous) functions by Lipschitz functions ?
3
votes
0answers
64 views

Replacing the “if x ≤ y, then x + z ≤ y + z” axiom in Reals.

How can I prove that we cannot (or maybe can) replace preservation of order under addition i.e. "If $x \leq y$, then $x + z \leq y + z$ with "if $0<x$ and $0<y$ , then $0<x+y$" in axioms ...
0
votes
1answer
20 views

Composition and Limits at Infinity

It is a well known result that if a function $f$ is continuous at $b$ and $\lim_{x\rightarrow a} g(x)=b$, then $\lim_{x\rightarrow a} f(g(x))=f(b)$. When does this hold at infinity? If $f$ is ...
1
vote
1answer
18 views

Books with exercises or problems on the space of functions of bounded variation

I am studying BV space (the space of functions of bounded variation) by using Evans & Gariepy's book. However, there are no exercises for BV space, and I have no idea where I can find some. ...
1
vote
4answers
57 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the ...
2
votes
3answers
44 views

Show that $f(x)=x/\sqrt{x^2+1}$ is a bijection of $\mathbb R$ onto $\{ y: -1<y<1\}$

I am looking for help in regard to a practice question about functions. The question is Show that a function $f$, defined by $f(x)=x/\sqrt{x^2+1}$ , $x \in \Bbb R$ is a bijection of $\Bbb R$ onto ...
0
votes
0answers
28 views

Find the point implied by intermediate value theorem

Consider a function $f(x)$ such that $f(0)=0$ and $$f'(x) = \frac{T-x}{T-f^{-1}(x)} + \frac{T-x}{S}$$ Then we can see that $f'(0)>1$ and $f'(T)=0$. Find $x$ such that $f'(x)=1$, in terms of the ...
4
votes
1answer
57 views

Characterization of measurability by closed sets.

If $E \subseteq \Bbb R$ is measurable, then for all $\epsilon > 0$, exists $F \subseteq \Bbb R$ closed such that $F \subseteq E$ and ${\frak m}^*(E \setminus F) < \epsilon$. I have already ...
0
votes
1answer
35 views

What is missing? (Rudin's Principles of Mathematical Analysis - Theorem 2.30)

Let us first give a definition: Definition Given a metric space $X$, and a subset $Y\subseteq X$, we say a subset $E$ of $Y$ is open relative to $Y$ if for each $p\in E$ there is an associated ...
3
votes
0answers
52 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) \end{equation} ...
0
votes
0answers
11 views

Identifying or bounding the zeros of the composition of two generating functions

Given two generating functions $$ G(a_n;x)=\sum_{n=0}^\infty a_nx^n \quad\text{ and }\quad H(b_n;x)=\sum_{n=0}^\infty b_nx^n, $$ what techniques are available for locating, or finding bounds on, the ...
1
vote
1answer
25 views

Question about prove of Absolute Convergence Test?

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I have a question about the proof of the Absolute Convergence Test (theorem 2.7.6 on page 65). The author states that ...
1
vote
0answers
21 views

taylor series expansion, derivatives not continuous

As a part of an excercise I am supposed to find the Taylor series expansion for $(1-t)^{\frac{1}{2}}$ on $[0,1]$. According to the remainder theorem: ...
2
votes
1answer
42 views

Convergence in measure and convergence in $L^p$

If $f_n$ is convergent to $f$ in measure and $\|f_{n}(x)\|_{L^{p}(\mathbb{R})}=\|f(x)\|_{L^{p}(\mathbb{R})}$. Does it implies that $f_n$ is convergent in $L^p$?
1
vote
2answers
39 views

Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
0
votes
1answer
31 views

Relation between limit of functions and sequences

I need to prove that the sequence $$ \frac{\sqrt{n}}{\log n}$$ diverges. I know that $$\lim \frac{\sqrt{x}}{\log x} = \infty$$. Is there any theorem that relations the limit of the function with the ...
1
vote
1answer
29 views

For every $n$ there exists $m$ such that $m/n$ is an upper bound but $(m-1)/n$ is not

This is a problem discussed in Analysis 1 by Terence Tao. $E$ is a non-empty set of Real numbers. $ n\geq 1$, $L,K$ are two integers such that $L<K$. Let $\frac{L}{n}$ is not an upper bound of ...
1
vote
1answer
44 views

Two sequences defined by recurrence relations satisfy $x_n/y_n<\sqrt{7}$ for all $n$

Let $(x_n)_{n\geq 1}$ and $(y_n)_{n\geq 1}$ be two sequences such that: $$x_{n+1}=x_n^2+1 \quad \text{ and } \quad y_{n+1}=x_n y_n$$ with $x_1=2$ and $y_1=1$ Prove that for all $n$ ...
5
votes
0answers
77 views

Show that it is an algebra.

This excercise is a little struggling for me. The part I need help with is showing that $D$ is closed under complements. Let $C$ denote the collection of all intervals on $\mathbb{R}$, including ...
1
vote
1answer
31 views

About composition of Holder functions.

Let $f,g$ be Holder continuous functions with respective exponents $\alpha, \beta \in (0,1)$. More precisely $f \in C^{\alpha}(\mathbb{R}^n;\mathbb{R}^n)$, $g\in C^{\beta}(\mathbb{R}^n,\mathbb{R})$. ...
-1
votes
0answers
23 views

Single formula sequence partitionining interval

So I have the real sequence for fixed $x\in \mathbb{R}$: $y_{j}(x)=\begin{cases}f_{j-1}(x) &\text{if } |f_{j-1}(x)|\leq |x_{j}|, \\ f_{j}(x) & \text{else}, \end{cases}$ where 1) ...
0
votes
1answer
31 views

A function relating $k$ and $j$, where $k=\max_{l\in \mathbb{N}}\sum_{i=0}^{l}2^{n-i}\leq j$ and $n= \lfloor \log_{2}j \rfloor$

Do you know any function that relates k and j, where $k=\max_{l\in \mathbb{N}}\sum_{i=0}^l 2^{n-i}\leq j$ and $n=\lfloor \log_2 j \rfloor$? So, say, for $j=3$: $n=1$ and $k=1$ because $3\geq ...
2
votes
4answers
73 views

For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
2
votes
0answers
40 views

simultaneous trigonometric equations

Consider the pair of simultaneous equations $p_5\cos(2\omega\tau)+p_4\omega\sin(2\omega\tau) = p_1\omega^2-p_3-p_6\cos(3\omega\tau) $, $p_4\omega\cos(2\omega\tau)-p_5\sin(2\omega\tau) = ...
1
vote
1answer
30 views

Proving IMVT using delta-epsilon

Let's assume $f(a)<0$ and $f(b)>0$. IMVT claims that there's $c\in(a,b)$ such that $f(c)=0$. The Proof: Consider $$A = \{ a\le x\le b : f(x) < 0 \}$$ That's a non-empty set and therefore, by ...
1
vote
1answer
59 views

If the integral $\int_0^\infty xf(x)\,dx$ converges, so does $\sum_{n=1}^{\infty}\int_0^{\infty}f(x+n)\, dx$

Let be $f:[0, \infty)\rightarrow [0,\infty)$ a measurable function such that $$\int_0^{\infty}x\cdot f(x)\,dx< \infty.$$ Show that $$\sum_{n=1}^{\infty}\int_0^{\infty}f(x+n)\, dx<\infty .$$ ...
2
votes
2answers
38 views

If $f(x)g(y)$ is a measurable function, and $f$, $g \in L^{1}(dm)$, does this imply $g(y - x) \in L^{1}(dm)$?

Question rephrased Suppose we are working in $(\mathbb{R}, \Sigma(m^{*}) \times \Sigma(m^{*}), m \times m)$ where $m$ is Lebesgue measure. Note that our $\sigma$-algebra is not necessarily complete. ...
0
votes
1answer
43 views

sigmal algebra and measure [on hold]

Let $\mu$ and $\lambda$ be two measures on a $\sigma$-algbra $\mathfrak{F}$ on $\Omega$, such that $\mu (A)=\lambda(A)$ for any $A\in \mathfrak C$, where $\mathfrak C\subset\mathfrak{F}$ is a ...
1
vote
2answers
43 views

Construction of a small but fat set? [duplicate]

Is it possible to find a subset $A$ of the real line $\mathbb R$ such that the Lebesgue measure of $A$ minus its interior is positive ?
1
vote
1answer
43 views

$x\cdot 0 \neq 0$ infinitely many zeroes on a finite interval

I was playing around with polynomials by looking how they behave when I move around their zero points. I decided to make a function which is zero on a finite interval, but since its a polynomial, it ...
2
votes
1answer
37 views

Prove $f$ isn't continuous at $\frac{1}{\pi}$

Let $f(x)=\left\lfloor {\sin {1 \over x}} \right\rfloor$ (meaning floor of $\sin x$). I need to prove that $f(x)$ isn't continuous at $x=\frac{1}{\pi}$. Proof: For a nehiborhood of $\frac{1}{\pi}$: ...