Can difference quotient sets be nowhere dense?
Let $f:\mathbb{R} \to \mathbb{R}$ be a function and consider the difference quotient set $$D_f = \left\{\frac{f(y) - f(x)}{y-x} : (x,y) \in \mathbb{R}^2, y > x\right\}$$Can $D_f$ be nowhere dense in...
View ArticleIVP with the Banach fixed point theorem: $y' = \sqrt{x} + \sqrt{|y|}$ and...
I need to use the Banach fixed point theorem to prove that $y' = \sqrt{x} + \sqrt{|y|}$ (for $x \geq 0$) with the initial condition $y(0)=0$ has a unique solution.First of all:\begin{equation}y' \geq...
View ArticleIs the Evaluation Map on Bounded Borel Measurable Functions Borel Measurable?
I am working with a set $I$, defined as the closed interval $[0,1]$, and a set $X$, which consists of all bounded Borel measurable functions defined on $[0,1]$. The uniform metric on $X$ is defined...
View ArticleIf f(x) and g(x) have equal value and the same derivative at x=0, do they...
In Wikipedia page on germ, it says thatGiven a point $x$ of a topological space $X$, and two maps $f,g\,:\,X\to Y$ (where $Y$ is any set), then $f$ and $g$ define the same germ at $x$ if there is a...
View ArticleLittle Hölder spaces - Reference request
The little Hölder space $c^\gamma(\mathbb{R}),$ for $0<\gamma<1,$ is usually defined in one of the following ways:$c^\gamma(\mathbb{R})$ is the clousure of $C^\infty(\mathbb{R})$ in the usual...
View ArticleWhy need the finiteness of $\mu(A)$ and $\mu(B)$ to define measurable...
In this errata of Real Analysis by Royden and PM Fitzpatrick, it corrects the definition of measurable rectangles to be $A\times B$ if $A\in\mathcal{A}, B\in\mathcal{B}$, and $\mu(A)$ and $\mu(B)$ are...
View ArticleBlumberg's theorem
The Blumberg theorem states that for any real function$𝑓: \mathbb{R}→\mathbb{R}$ there is a dense subset $𝐷$ of $\mathbb{R}$ such that the restriction of $𝑓$ to $D$ is continuous.In the original paper...
View ArticleDistribution and convolutions [closed]
Let $f(x,y)=y^2\chi_E(x,y)$ with $E= \{ (x,y) \in \mathbb{R}^2 s.t. x\ge 0\}$, let $B$ the unit ball in $\mathbb{R}^2$ and let $\{\rho_\epsilon \} $ standard mollifier.I should compute...
View ArticleFormulation of Axioms about Landau Big $O$ and Small $o$ Notation Arithmetics
My question is, how to evaluate and prove the situation when we have something like , say $o[\frac1{2x}+o(\frac 1x)]=o[\frac 1x]$.In which I encountered this case while calculating the...
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Are the two notions of subgaussian variable equivalent?
I have seen the following two definitions of subgaussian so far:Definition 1:Definition 2:It seems like the only major difference between the two is the constant $2$ in the front. Is there a way to...
View Articlewhat is Q-regular banach space? [closed]
I have been reading a research article about banach space.They mentioned about Q regular
View ArticleIf $ \lim_{n\to \infty} (a_n - b_n) = 0$, with $\{a_n\}$ converging, then...
Here is a question I am working on: Let $\{a_n\}$ be a sequence with limit $L$. Suppose that $\{b_n\}$ is a sequence such that for some positive integer $N$ we have $b_n$ = $a_n$ for every $n\ge N$....
View ArticleAdding increasing function to an integral
Suppose the following holds for a continuous f(x) defined on the real line: $$\int_{-\infty}^{+\infty}f(x)dx\geq 0$$where $f(x)$ can take both positive or negative values.Suppose there is a function...
View ArticleCurvature measure for convex functions in $\mathbb R^d$
TLDR: convex functions in high dimensions are much weirder than in 1D: do you please have insights or references to share?Let $c(x)$ be a convex function on the real line.There is an obvious measure of...
View ArticleQuestion on dominating convergence theorem
On $[0,1]$ we have sequence of non-negative continuous functions {${f_n}$ } converging pointwise to $f(x)$,and $f_n(x)\leq f(x)~\forall~x\in[0,1].$Is this condition enough to conclude the...
View ArticleIs it possible to evaluate...
I am curious to know if it is possible to evaluate the following integral:$$\int_{-\infty}^{\infty}e^{-x^2}\operatorname{sech}^2\left(\frac{x}{2}\right)\,dx$$Here is the context:Someone had asked me...
View ArticleIs Darboux's theorem an extension of intermediate value theorem
Darboux theorem states that the intermediate value property of every bounded derivative on $[a,b]$. Since every continuous function on closed interval $[a,b]$ must be a derivative of some functions, it...
View ArticleSum that converges to arbitrary $s$
I am just curious about this problem: suppose I know that $\sum_{n=1}^{\infty} a_n = L$ and $a_n>0$ for all $n$. Does there exist a subsequence $\{a_{n_k}\}$ such that $\sum_{k=1}^{\infty}a_{n_k}=x$...
View ArticleDifferentiation under the sign of double integral
Consider the double integral $$I(a)=\int_0^1\int_0^1 f(x,y,a)dxdy$$What are the conditions needed so that we can differentiate under the integral sign of this double integral? That is when do we have...
View Articleconvergence $f\ast \rho_{\epsilon}\to f$ when $\epsilon \to 0$
Let $(\rho_{\epsilon})_{\epsilon\ge 0}$ be mollifiers, i.e. $C^{\infty}(\mathbb{R}^n)$ with $\rho_{\epsilon}\ge 0, \int \rho_{\epsilon} = 1$ and $supp(\rho_{\epsilon})\subset B(0,\epsilon)$. I want to...
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