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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On the existence of a weight function making sequence of integration preserving limit

The problem goes as follows: Let $f_n$ be strictly positive Lebesgue measurable function defined on $[0,\infty)$ satisfying $$\lim_{n \to \infty} \int_0^\infty f_n(x)\ dx=0$$ then show that there ...
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174 views

Simple problem of a differentiable function

Please, can somebody help me with this problem? I tried to use the Mean Value Theorem, but couldn't solve it. Let $g: [a,b]\rightarrow\mathbb{R}$ a differentiable function on $[a,b]$. If ...
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330 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 ...
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293 views

Bartle or Cohn as a first text in Measure Theory.

Consider the following texts: (a) Robert Bartle - Elements of Integration and Lebesgue Measure (b) Donald Cohn - Measure Theory. Which text would you recommend for a student with only a modest ...
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617 views

Strict Inequality in Rudin's Proof of the Riesz Representation Theorem

In Rudin's proof of the Riesz Representation Theorem (step ten), he proves that $$\Lambda h_i \leq \mu(V_i) < \mu(E_i) + \epsilon/n , \quad \mu(K) \leq \sum\limits_{1 \leq i \leq n} \Lambda h_i.$$ ...
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139 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ ...
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74 views

Constructivism implied or not

Let me take up some details in the answer of another question. Submitted by user hyg17: Heading: All real numbers can be expressed as a limit of rational numbers? The question was: Let $C$ be a set ...
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203 views

Lebesgue measure as a fixpoint: change of variables formulas

This question is inspired by several others on a similar topic: see e.g. this one and a sequence of linked questions. Let us so far focus on $\Bbb R^n$ endowed with standard Borel structure. For any ...
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172 views

Stokes' Theorem problem

Let $M \subset \mathbf{R}^n$ be oriented compact smooth $k$-manifold and $\alpha$ be a $C^1$ diferential $(k-1)$-form defined in a neighborhood of M. Use Stokes' theorem to prove that \begin{align*} ...
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85 views

Possible error in Wood's report on polylogarithms

I'm studying the article by Wood, D.C. "The Computation of Polylogarithms. Technical Report 15-92*" PS (it is remarkably poorly translated from latex to ps). It is listed in the literature section on ...
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82 views

$f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$

Consider the equation: $f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$, for $a \geq 0$ and $b \geq 0$. Is my understanding that this simple functional equation is important in analysis. Can ...
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2k views

Proving rigorously the supremum of a set.

Suppose $A \subset \mathbb{R} \neq \emptyset$. Let $A = [\,0,2).\,\,$ Prove that $\sup A = 2$ Attempt: $A$ is the half open interval $[\,0,2)$ and so all the $x_i \in A$ look like $0 \leq x_i < ...
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81 views

Partition of $\mathbb{R}$ into nullset and 1st category set

Let $\{a_i\}_{i=1}^{\infty}$ be an enumeration of the rationals, $\mathbb{Q}$. Let $I_{ij}$ be the open interval centered at $a_i$ and having length $1/2^{i+j}$, and define $G_j = \cup_{i=1}^\infty ...
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576 views

Infinite self-convolution for a function

I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times. So consider a generic function $f : ...
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71 views

$x\mapsto f(x^{1/p})$ is smooth if and only if $f^{(n)}(0)=0$ whenever $p\nmid n$

This is a slight generalization of something I got stuck on when trying to do Problem 2-5 from Introduction to Smooth Manifolds by John M. Lee (which uses the case $p=3$). Let $p\ge 1$ be an ...
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208 views

interchange limits (not a sequence of function)

I am aware of a theorem in analysis regarding the result that $$\lim_{t\rightarrow x} \lim_{n\rightarrow \infty} f_n(t) = \lim_{n\rightarrow \infty} \lim_{x\rightarrow t} f_n(t).$$ My question is ...
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297 views

An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.

We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
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233 views

Second derivative of a vector field

I wonder how to treat the "second derivative" of a vector field. For example, imagine we have a vector field $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Then we evaluate the derivative at two points ...
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433 views

Help me correct these properties of : $f_{n}(x)= nx(1-x)^{n}$? Is there maybe a typo in the sequence?

Examine the sequence of functions $(f_n)_{n\in \mathbb{N}}$ on $x\in[0,1]$: $$f_{n}(x)= nx(1-x)^{n}$$ Does $(f_n)_{n\in \mathbb{N}}$ converge pointwise or uniform? I will show that it does ...
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89 views

Use Lagrange's method to find the maximum value

Use Lagrange's method to find the maximum value of $\langle A\mathbf{x},\mathbf{x}\rangle$ subject to condition $\langle \mathbf{x},\mathbf{x}\rangle=1$ and $\langle \mathbf{u}_1,\mathbf{x}\rangle =0$ ...
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88 views

The Limit of the Following Derivative

Suppose you have two functions $F$ and $G$ with the following properties. $G(0)=F(0)=0, G'>0, G''<0, F'>0, F''<0 $ and also $\lim_{x\to0} F'(x)=\infty, \lim_{x\to\infty} F'(x)=0, ...
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168 views

Constructing the support of a Borel measure

From Rudin, Real and Complex Analysis, Chapter 8, Problem 7, 1st Edition. Suppose $E$ is a compact set in $\mathbb{R}^{k}$ without isolated points. Show that $E$ is the support of a continuous ...
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129 views

When does $|f*g|_{p}=|f|_{1}|g|_{p}$?

From Rudin, Real and Complex Analysis, 1st edition, Chapter 7, Problem 4 Suppose $1\le p\le \infty$, $f\in L^{1}(\mathbb{R}^{1})$, $g\in L^{p}(\mathbb{R}^{1})$. Show that the the integral defining ...
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161 views

Inexact Newton method.

Let's a nonlinear function $ f:[-1,+1]^N\subset\mathbb{R}^N\to\mathbb{R}^N,\; N\in\mathbb{N}, $ such that the the sequence generated by the method of Newton-Raphson $$ x_{n+1}=x_n-[Df(x_n)]^{-1}\cdot ...
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156 views

Riemann Integrable $f$ and Real Analysis Proofs

I am solving old comprehensive real analysis exams and there are two questions that I can not be sure, If $f$ is Riemann integrable then $|f|^r$ is Riemann integrable for any $r>0$.( True or ...
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158 views

Functions from the Cantor set

Consider the Cantor set $\Delta\subset [0,1]$. Let $f\colon \Delta\to [0,\infty)$ be a continuous injection. Must $f$ be monotone on some uncountable closed subset of $\Delta$? Note that that van der ...
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1k views

prove that every continuous function is integrable

Can someone tell me whether this is correct thank you! We know that if a function f is continuous on $[a,b]$, a closed finite interval, then f is uniformly continuous on that interval. This means ...
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119 views

A question about functions in $L^p(E)$

I've been working on a problem. I found a couple of related questions on the site, but I was having a little bit of trouble clarifying everything. The goal is to show that given $f\notin L^p(E)$, ...
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184 views

Equivalent definition of lower semi-continuity

I have some problems about solving an exercise: Prove that one function $f \colon [a,b] \to \mathbb{R}$ is lower semi-continuous if and only if, for all $x \in [a,b]$, we have $$f(x)=\sup \{g(x) ...
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235 views

Prove that liminf of functions is semicontinuous

I recall that, if $\psi:\Bbb{R}\longrightarrow \Bbb{R}$ is a function defined over $\Bbb R$ with euclidean topology, we have $\liminf\limits_{y\to x} \psi(y) = \sup\limits_{U\in \mathscr{U}_x} ...
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137 views

How many points does one need for an epsilon-net

Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
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974 views

Closure, Interior, and Boundary of Jordan Measurable Sets.

This question has a number of parts. Let $E\subset\mathbb{R}^{d}$ be a bounded subset. (1) Show that $m^{\star,(J)}(E)=m^{\star,(J)}(\bar{E})$ (closure) (2) Show that ...
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131 views

$C(K,R)$ continuous functions on compact metric spaces

How to do this problem? Question: let $(K,d)$ be a compact metric space and $A$ is contained in $C(K,R)$, the set of all continuous functions from $K$ to $\mathbb R$, an algebra that separates the ...
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166 views

Help with removing singularities involving $ \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}$

This post can be thought of as the prototype proof and the motivation for the question posted here Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion ...
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160 views

Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral $$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$ In the investigation of ...
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373 views

Interchanging the order of limits

Would you advise me on the references of Pringsheim Convergence about interchanging the order of limits? Where can I find the most general statement?
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757 views

First order variation and total variation of a function/stochastic process

The notions of first-order variation and total variation of a function or a stochastic process are equated in this book. However, I found their definitions different from two other sources: In ...
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136 views

Arclength integral

Suppose that $f: [a,b] \rightarrow \mathrm{R}^n$ is continuous with a derivative $f'$ whose norm is Riemann-integrable. To demonstrate the arclength integral formula, I'm trying to prove that, for ...
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439 views

Convergent sum with primes

If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
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2k views

Derivative of a differentiable everywhere function $f\colon \mathbb{R}\to\mathbb{R}$ is Borel measurable?

So my homework problem is to show that if a function $f: \mathbb{R} \to \mathbb{R}$ is differentiable everywhere, then its derivative $f'$ is Borel measurable. What I have for something to be ...
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171 views

Uniform convergence of piecewise linear interpolations

Let $$X^k (t) := X^0 (t+t_k) - X^0 (t_k)$$ where $X^0(t)$ is the piecewise linear interpolation of $X^0(t_k) \equiv X_k=\sum_{i=0}^{k-1} a_i b_i$ with interpolation intervals $a_k$. ...
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213 views

Equivalence of two sequences

I'm having some trouble showing that two things I really want to be the same are in fact the same. I want to show that these two sequences are, in fact, the same thing: $$a_0=1,a_1=-1, ...
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439 views

How to construct a vector space and compute basis?

My professor demonstrated that in vector calculus that you can construct basis vectors for one, two, and three forms using the vectors $dx$, $dx$ and $dy$, as well as $dx \wedge dy$, $dy \wedge dz$, ...
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148 views

Reference for theory of non-uniformly convergent sequences of functions

I have been thinking about some conjectures of mine regarding non-uniformly convergent sequences of functions. I'm happy to discuss my work so far, if anyone wants to know, but for brevity let me ...
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15 views

Integral Asymptotics for inhomogenous phase

I'm looking for asymptotics for an integral of the form: $$F(n):=\int_{1/2-i\infty}^{1/2+i\infty} e^{\phi(n,z)}dz$$ where $\phi(n,z)=(n-n^3)\log(1-z)+n^2\log(1+z)-n\log(z)$. One can solve for the ...
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54 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
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110 views

A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to ...
3
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30 views

Spectra of periodic Schrödinger equations

This question might be a little bit physics-related, but I kind of have a deep interest to ask this here, cause I would like to get an idea of the Mathematics behind the things I just covered in my ...
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25 views

The unit ball in $L^p$

Let $X$ Consist of two points $a$ and $b$, put $\mu(\{a\})=\mu(\{b\})=1/2$, and let $L^p(\mu)$ be the resulting real $L^p$ space. $\textbf{Identify each real function $f$ on $X $ with the point ...
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30 views

Approximating higher dimensional Lipschitz maps with differentiable functions

Suppose $f:[0,1]^n \to {\mathbb R}$ is Lipschitz. It is known that $f$ is differentiable almost everywhere. Can we construct a sequence $f_n$ of continuously differentiable functions such that $\lim ...