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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3
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24 views

Check my proof: Prove that if $f$ is defined as having a positive disntinuity at $c$ and $0$ otherwise on [a,b], it is Darboux integrable

Prove that if $f$ is defined as having a positive discontinuity at $c$ and $0$ otherwise on [a,b], it is Darboux integrable and its integral is 0. $\forall \epsilon>0,$ choose ...
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77 views

Why do people apply Fubini-Tonelli theorem so easily?

I'm reading a text "Lebesgue Integration - Frank jones" from which i got recommended here, stackexchage. This text seemingly covers various topics on measure theory, but i think that's it. This text ...
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47 views

Convex Functions are Continuous

In W. Rudin's Principles of Mathematical Analysis, we read in Chapter 4 that real-valued functions defined on an open interval $(a,b)\subseteq\Bbb R$ are continuous (specifically, Exercise 23). I am ...
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28 views

Intution, Figure. Negation of Continuity and Uniform Continuity (S.A. pp 117 T4.4.6)

Every time I need negation, I have to write out all the logical symbols to negate manually. I know how to determine these negations myself. But I want to compehend intuition or figure like ...
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47 views

Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there ...
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168 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
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91 views

My proof of Bolzano's theorem

Before I read the proof of Bolzano's theorem from my Calculus book, I've tried to prove it myself. I will use the following lemma and the least upper bound axiom. [Lemma: Sign-preserving property of ...
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57 views

Definition of logarithmic capacity

In the definition of logarithmic capacity of a compact set $E$ in the plane, the Robin constant is defined to be $V(E)=inf\int_E\int_E log\frac{1}{|z-w|} d\mu(z)d\mu(w)$ where $inf$ is taken over all ...
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35 views

Is this a decomposition of the same function?

Let's say we have some integral, such that for a particular function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ $$\int_{\mathbb{R}^{n-m}} \int_{\mathbb{R}^m}f^+ - ...
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61 views

Can we have an isometric embedding of this metric space into an Hilbert space?

A metric space (from this Q&A), is defined below. I'd like to know if its possible to have an isometric embedding of this metric space into an hilbert space? As per Schoenberg theorem $-d^2(x,y)$ ...
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31 views

Analog of Fubini's theorem for Borel functions

Suppose that $f\colon [0,a]\times [0,b] \to \mathbb R$ is a Borel function, where $a>0$ and $b>0$. Let $\mathscr F$ denote the space of Borel functions $[0,b]\to \mathbb R$, and let us endow ...
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198 views

special combinatorial sums

I know several of combinatorial sums, which is equal to 0. For example, $$ \sum_{0 \leq j \leq d} (-1)^j {d \choose j}=0$$ $$ \sum_{0 \leq j \leq \frac{d}{2}} (-1)^j {d \choose j}(d-2j)^\alpha=0, \, ...
3
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60 views

Solve $\frac{x}{\sin x}+\frac{1}{\cos x}-\pi=0$

Solve $\frac{x}{\sin x}+\frac{1}{\cos x}-\pi=0$ I'm aware there will be an infinite number of answers. Can you solve this exactly or does it have to be done numerically?
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53 views

Inequality $\sum_{k=1}^{n}\frac{k}{a_{1}+\cdots+a_{k}}\le\left(2-\frac{7\ln{2}}{8\ln{n}}\right)\sum_{k=1}^{n}\frac{1}{a_{k}}$

$n\geq2$ is a integer, $a_n\gt0, n=1,2,\dotsc$, then $$\sum_{k=1}^{n}\dfrac{k}{a_{1}+a_{2}+\cdots+a_{k}}\le\left(2-\dfrac{7\ln{2}}{8\ln{n}}\right)\sum_{k=1}^{n}\dfrac{1}{a_{k}}$$ I don't know how to ...
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129 views
+50

Cantor set exercise

This is an exercise from Abbott's real analysis book. It's exercise 3.4.4.(b) on page 93. I couldn't find a definition of ''dimension'' in the book. The only thing I could find is something on page ...
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57 views

Does the implicit function theorem imply Peano existence theorem

In The implicit function theorem written by Krantz & Parks, it's said that the implicit function theorem implies the following existence theorem of ODE: Theorem 4.1.1 If $F(t,x)$, ...
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40 views

Choosing an appropriate complete orthogonal basis

I have a function $f(x)$ which I want to represent as the sum over some complete orthogonal basis $\phi_i$ such that: $$ f(x) = \sum_{i} c_i \phi_i(x) $$ Where the $\phi_i$ are orthogonal with ...
3
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60 views

Is $f'$ continuous on $[a,b]$?

If $f$ is continuous on a compact interval $[a,b]$ and has a continuous derivative does it mean that $f'$ is continuous on $[a,b]$ or $(a,b)$?
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37 views

Norms on $\mathbb{Q}$

So with respect to the metric $d(x,y)=|x-y|$ induced by the standard absolute value, the real numbers can be constructed as a completion of $\mathbb{Q}$. With respect to the metric $d_p(x,y)=|x-y|_p$ ...
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98 views

Volume of “deformed torus”

I'm trying to find explicit form of volume of "deformed torus": Suppose we have a curve $\gamma(t)$ in $\mathbb{R}^n$, $t\in[0,1]$. The curve closed and smooth : ...
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29 views

Continuous $L^2$ function has finite sum at integer points?

Let $f\in L^2(\mathbb{R})$ be a continuous function. Is it true that $\sum_{n=1}^\infty |f(n)|^2$ is finite? If the continuity condition is dropped, the statement is not true, because $f(n)$ could ...
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46 views

Cantor measure not supported outside Cantor set

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be the Cantor function, which is continuous. Define the Cantor measure by $\mu((a,b))=f(b)-f(a)$ for $-\infty<a<b<\infty$. Denote by $C$ the standard ...
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67 views

lim sup intuition for a sequence of sets

lim sup of a sequence of sets $(E_n)$ is defined as $$\bigcap_{n = 1}^{\infty}\bigcup_{k = n}^{\infty} E_k$$ and this means that an element ins lim sup $E_n$ is a member of infinitely many of the ...
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70 views

Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
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110 views

Applying Dominated Convergence Theorem to solve $\lim_{n \to \infty}\int_0^n(1-\frac{x}{n})^ne^{-x}dx$

$$\lim_{n \to \infty}\int_0^n(1-\frac{x}{n})^ne^{-x}dx$$ I am going to use the Dominated Convergence Theorem to solve this. $$\lim_{n \to \infty}\int_0^{\infty}\chi_{[0, ...
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36 views

representation of points of continuity of a function $f :\mathbb{R}\rightarrow \mathbb{R}$

Question is : Suppose $f$ is continuous at $x\in \mathbb{R}$ we need : for given $\epsilon >0 $ existence of $\delta > 0$ such that $|x-y|< \delta$ implies $|f(x)-f(y)|< \epsilon$ ...
3
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23 views

Showing measures of random variables are equal

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given random variables $f,g\colon X\rightarrow\mathbb{R}$, suppose I want to show that ...
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38 views

Zero arithmetic mean: bound on Abel sum?

Let $(a_0, a_1, a_2, \ldots)$ be a bounded sequence in $\mathbb{R}$ with arithmetic means \begin{equation} \frac{a_0 + a_1 + \cdots + a_{n-1}}{n} \end{equation} converging to zero as $n \to \infty$. ...
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74 views

A question on mathematical writing.

One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal ...
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34 views

Lie differentiation operation for manifolds

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. Let ...
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54 views

Direct proof that $\sum_{n>0}x^n/n!=\lim_{n\rightarrow\infty}(1+x/n)^n$

Is there a direct proof that $\sum_{0\leq n}x^n/n!=\lim_{n\rightarrow\infty}(1+x/n)^n$ ? We dont know what logarithms or exponentials are.
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52 views

Recovery of Bandlimited Signals

Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) ...
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53 views

Push-forward vector space calculation

Let $v$ be the vector field $x_1\dfrac{\partial}{\partial x_2}-x_2\dfrac{\partial}{\partial x_1}$, and let $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be the diffeomorphism $(x_1,x_2)\rightarrow ...
3
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64 views

For the sequence $u_n$, $u_n \to +\infty \iff \frac{1}{u_n} \to 0$

Let $u=(u_n)_{n \in \mathbb{N}}$ be a sequence such that $u_n \neq 0$, $u_n \to +\infty$, for $ n \to +\infty$. Proof that $u_n \to + \infty , ( n \to +\infty) \iff \left(( \exists n_0 , ...
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57 views

An orthonormal basis for a Hilbert space

Can anyone give me some hint on the following problem without using any knowledge about complex analysis or Fourier analysis? Thanks a lot! Consider the Hilbert space $$\mathscr{H}:=\bigg\{f\text{ ...
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48 views

Basis in $L^2([-\pi,\pi])$

Consider the space $L^2([-\pi,\pi])$. Show that the functions $f_0(x)=1,f_1(x)=x,f_2(x)=x^2,\ldots $ form a basis. The functions are linearly independent (no linear combination adds up to zero). But ...
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102 views

Integrating the exponential of a complex quadratic matrix

Problem statement I'm trying to do a discretized path integral/functional integral. The integral that I'm stuck with is of the form $$ \int_{-\infty}^{+\infty} \mathrm{d}^3\vec{x}_1\, ...
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122 views

Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
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56 views

Why is completeness of measures important?

Can someone give an example why a non-complete measure is a problem? Or what are some theorems that require a measure to be complete?
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58 views

Decimal system history

Today, "numbers" usually refer to real numbers and are most commonly conceptualized as consisting of all possible infinite decimal expansions (or binary expansions, etc). When did this way of thinking ...
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113 views

If $f$ is integrable then so is $1/f$

Let $f$ be an integrable function on $[a,b]$ with $|f| \geq p > 0 $ for $a \leq x \leq b$. Show that $\frac{1}{f}$ is also integrable on $[a,b]$. I was told the Cauchy-Schwarz Inequality might be ...
3
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57 views

convergence of series implies convergence of coefficients

Is it true that $$\sum_{i=0}^\infty a_{i_n} y^i \rightarrow \sum_{i=0}^\infty a_{i} y^i \quad \forall y \in [0,1]$$ implies $$a_{i_n} \to_{n \to \infty} a_{i} \quad \forall i$$ where $0 \leq ...
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239 views

The product of a uniformly continuous function and a bounded continuous function is uniformly continuous

Suppose we have a bounded continuous function $f(x)$ on some interval (a,b). Suppose we also have an function $g(x)$ that is uniformly continuous on the same interval (a,b). Then, is the product ...
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129 views

derivative of limit function vs limit of derivatives

Suppose that we have a sequence of differentiable functions $f_n:\mathbb{R} \rightarrow \mathbb{R}$ such that $f_n$ converges to some function $f$. Then it is not necessary that the sequence of ...
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78 views

$|f(t) - f(s) |\leq \int_s^t g $ then $f(t) - f(s) = \int_s^t h.$

Let $f : [0,1] \rightarrow [0, + \infty)$. If there exists $g \in L^1([0,1]) $ s.t. for every $t,s \in [0,1]$ holds $$ |f(t) - f(s)| \leq \int_s^t g(u) \, du \quad (t>s),$$ then there ...
3
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46 views

The infinite sum of functions

Can anyone give me some hints about the following problem? Many thanks! Let $\{f_n\}_{n\ge 0}$ be a sequence of integrable functions where each $f_n: (E, \mathcal{M},\mu)\to \overline{\mathbb{R}}$ ...
3
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42 views

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be continuous. $A$ is in the Borel subset of $\mathbb{R}$. Show that the $f^{-1}(A)$ is Borel.

Here is my attempt: First, prove that any open set E has the property that $f^{-1}(E)$ is Borel. This is true because the inverse image of open set is open under continuous functions. Next, prove ...
3
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79 views

Lebesgue outer measure, real analysis

Let $E$ have finite outer measure. Show that $E$ is measurable if and only if for each open, bounded interval $(a,b)$, $(b-a)=m^*((a,b) \cap E)+ m^* ((a,b) \cap E^c)$. The forward direction ...
3
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0answers
35 views

How to find minimum$(|a_n|+|a_{n-1}|+\cdots+|a_0|)$ if $f(x)=0$ has at least one root $x\in (0,r)$?

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0,a_i\in\mathbb Z,n\in\mathbb N.$ How to find minimum$(|a_n|+|a_{n-1}|+\cdots+|a_0|)$ if $f(x)=0$ has at least one root $x\in (0,r)$? (For example, ...
3
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64 views

$C^1$ map to greater dimension has measure zero

Let $f:\mathbb{R}^{n-1}\rightarrow\mathbb{R}^{n}$ be a $C^1$ map. Prove that the image of $f$ is a set of measure zero. I know the following theorem: Suppose $A\in\mathbb{R}^n$ is open and let ...