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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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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144 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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123 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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156 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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152 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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157 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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112 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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172 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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225 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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131 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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915 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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617 views

Characteristic functions based proof problem.

I am trying to show that if $T$ be a closed bounded interval and $E$ a measurable subset of $T$. Let $\epsilon >0$, then there is a step function $h$ on $T$ and a measurable subset $F$ of $T$ for ...
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258 views

What are the conditions sufficient and necessary on $g(t)$ for the Dirichlet integral to be equal to $\frac{\pi}{2} g(0+)$?

Dirichlet Integral of a function $g\colon \mathbb{R} \to \mathbb{R}$ is defined as $$ DI(\alpha) = \int_0^{\delta} g(t) \frac{\sin(\alpha t)}{t} dt$$ assume $\alpha \in \mathbb{N}$ For the equality ...
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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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159 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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359 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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717 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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132 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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Show that a differentiable everywhere function $f\colon \mathbb{R}\to\mathbb{R}$ has a derivative which 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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168 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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431 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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134 views

When is the $n$th term sufficient to guarantee convergence of a series

If $K$ is a field complete with respect to a non-archimedean absolute value, then the $n$th term test (checking whether the $n$th term of a series goes to zero) is sufficient to check convergence of a ...
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+50

Dimension score for metric spaces $\mathbb{R}^n$

Given any metric space $(X,d)$, define its score $S(X)$ to be the smallest value of $k$ such that for every $x\in X$ and $r>0$, the ball $B(X,r)$ is covered by at most $2^k$ balls of radius $r/2$. ...
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44 views

Do these limits commute?

Given a sequence of functions $f_{n,m}:\mathbb{R}^{n} \to \mathbb{R}$, suppose that $$\displaystyle lim_{m} f_{n,m}(x)$$ exists almost everywhere (for any fixed n) and also suppose that ...
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Finding disjoint intervals from Cantor Set

Consider $C$ the classic Cantor ternary set in $[0,1]$. I am interested in the following problem: Find the largest constant $0<k<1$ such that it is true that any interval $[a,b] \subseteq ...
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63 views

Nets and convergence of Riemann sum

STATEMENT: Let $[a,b]$ be a finite interval in $\mathbb{R}$. Let $\Lambda$ be the collection of all finite subsets of $[a,b]$ that contain $a$ and $b$. When orderedf by inclusion, it is a directed ...
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33 views

Does $L^{1}$ convergence implies almost everywhere cesaro convergence?

Let $X$ be a compact metric space, $\mu$ a Borel measure and $f_{n} \in L^{1}(X,\mu)$ continuous functions such that $f_{n}(x) \overset {L^{1}} \rightarrow 0 $. Now can we deduce that ...
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41 views

Equivalence of Lebesgue integral definitions

I'm currently enrolled in a course in integration and functional analysis following Avner Friedman's Foundations of Modern Analysis. However, I noticed that his definition of the Lebesgue integral is ...
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57 views

Can $\|f\|_p\to\infty$ arbitrarily slowly? (Looking for hints.)

Given $f$ is Lebesgue measurable on $(0,1)$ and not essentially bounded, is it true that to every positive function $\Phi$ on $(0,\infty)$ such that $\Phi(p)\to\infty$ as $p\to\infty$ one can find an ...
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Riemann Mapping Theorem for Homeomorphisms

We know by Riemann mapping theorem that for every connected, simply connected proper open subset $U$ of the complex plane, there exists a biholomorphic map $\phi:U\to \mathbb{D}$ (Where by ...
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Finding the general solution to system of linear equations: $y' = 2y,y''=4y-y'$

Question: I want to find the general solution to the following system: $\begin{pmatrix} \dot{y}_1 \\ \dot{y}_2 \end{pmatrix} = \begin{pmatrix} 2&0\\4&-1 ...
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73 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
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15 views

Show that if $X_\alpha$ is Hausdorff for all $\alpha$, then $\prod X_\alpha$ is Hausdorff under the box and product topologies.

Can someone please verify my proof? I am aware that there is a similar question posted elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is not homework. Show that if ...
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Compactness and Lipschitz functions

I am very stumped by this question: Suppose (K, d) is a compact metric space. Let f be any function, f: K $\rightarrow \mathbb{C}$, not necessarily continuous. Prove that for any $\epsilon > ...
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93 views

Convergence in probability and convergence of Cesaro means

Consider the random variable $X_n$, not necessarily iid. If $X_n\rightarrow 0$ almost surely, then the Cesaro means $\frac 1n\sum_{k=1}^nX_n$ converge almost surely to 0. This cannot be weakened to ...
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42 views

Check whether $f$ is continuous on $\Bbb{R}$

Let $\Bbb{Q}=\{q_1,q_2,q_3,\cdots\}$ be enumeration of rationals. Define $$f(x)=\displaystyle \sum_{q_i<x}2^{-i}$$ Investigate whether $f$ is continuous on $\Bbb{R}$. And is $f$ differentiable ...
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Showing equality of sets in $C[a,b]$

The exercise states: Let $a,b\in\mathbb{R}$, $a<b$ and let $(C[a,b],\Vert\cdot\Vert)$ denote the vector space of continuous real functions on $[a,b]$ endowed with the uniform norm. Let ...
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Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$.

Let $I = [a, b], E \subset I, m(E) = 0$ (but $E$ not empty). Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$. I am ...
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Let $\eta (x)=\int_0^\infty e^{at}\xi(\phi_t(x)) dt$ then $\eta$ is a $C^1$ function

Consider the following problem. Suppose that $a>0, r >0$ and $\xi:\mathbb R \to [o,\infty)$ is a $C^2$ which vanishes in the complement of the interval $(-r,r)$. Also suppose that ...
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85 views

Squeeze Theorem: Finding the limit of a trig function

I'm stuck on finding the limit of a complex fraction/trig function. Could someone please assist, or point out where I'm going wrong? Determine $$\lim\limits_{x \to 0} ...
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50 views

A sequence of continuous functions on $\mathbb{R}$ pointwise bounded at rationals

I made up my own question: If $\{f_n\}$ is a sequence of continuous functions on $\mathbb{R}$ pointwise bounded at rationals, then there exists a convergent subsequence $f_{n_k}\to f$ on ...
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48 views

Simplifying an integral involving Gaussian PDF

Let $\phi(x)$ be the standard Gaussian probability density function and $1<Y<2$.Consider the integral $$ \int_{x=0}^\infty \int_{y=0}^\infty ...
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Question about the definition of $\limsup$ and $\liminf$ on real valued functions

I understood the definition of $\liminf$ and $\limsup$ for sequences well. But there is a little bit of confusion when it comes to functions. If I did learn it correctly, the definition of ...
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74 views

Distinction between nowhere monotone and nowhere differentiable

It is known that all functions that are continuous and nowhere differentiable are also nowhere monotone but that there is a function that is everywhere differentiable but nowhere monotone. I have ...
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60 views

Integral substitution paradox

Assume $f \in L^+(\mathbb{R})$ and $x>0$. Consider the integral $$ \int_0^\infty \frac{f\left(\frac{x}{y}\right)}{y} \: dy. $$ I am trying to make the substitution $u=x/y.$ I seem to get $$ ...
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Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...