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Is there a continuity between sublevel sets of a convex function?
this is a bit of an open ended question.I have been thinking about the relationship between convexity and continuity of sublevel sets but couldn't find much about it. So I appreciate any thoughts,...
View ArticleLimit value depending on a function's asymptotics
I want to compute the limit of the following function when $n\to\infty$:$$f(n)=\frac{2 n \left(1-\frac{1}{n}\right)^{g(n)}}{2-3 n \ln \left(1-\left(1-\frac{1}{n}\right)^{g(n)}\right)}$$It depends on a...
View ArticleProperties of a $C^1$-function $f: [0,1] \to \mathbb{R}$
Assume we have a $C^1$-function $f: [0,1] \to \mathbb{R}$ with the following properties:$f(0)<0$ and $f(1)>0$I was asking myself if it is the case that there exists a point $x_0 \in (0,1)$with...
View ArticleIs there a result allowing to seperate two connected components
Let $X$ be a subset of $\mathbb{R}^n$ such that$$X = A \cup B,$$with $A\cap B = \emptyset$ and $A$ and $B$ are two connected non empty subsets of $\mathbb{R}^n$. Is there a result allowing to prove...
View Articlesaturated measure Folland exercise 1.3.16c trouble with a helpful Lemma proof
I'm solving Folland's Real Analysis' exercise 16 of session 1.3 and found some trouble. I found this question and have been able to follow the top answer, but there's a part that doesn't present much...
View ArticleBoundedness of the modulus of continuity
According to Wikipedia, the modulus of continuity is "used to measure quantitatively the uniform continuity of functions", and is defined as follows:For a function $f: I \rightarrow \mathbf{R}$ admits...
View Articlehow to prove the basic property of the Bochner integral in terms of the...
Definition A function $f: E \rightarrow \mathbb{X}$ is called simple if it can be represented as$$f(\omega)=\sum_{i=1}^k I_{E_i}(\omega) g_i$$for some finite $k, E_i \in \mathscr{B}$ and $g_i \in...
View ArticleEven assuming AC, how could one ever "choose" a unique, non-algorithmically...
Even assuming AC, how could one ever "choose" a unique, non-algorithmically specifiable, [like $\pi$, $e$, & $\varphi$ are] transcendental real number from, say, I?This seems like a fairly simple...
View ArticleThe proof of existence of an infinite sequence of disjoint sets.
Here is the question I am asking about:Let $\mathcal{M}$ be an infinite $\sigma$-algebra.\a. $\mathcal{M}$ contains an infinite sequence of disjoint sets.\Here are many links I found here for the...
View Article$\operatorname{cov}(\mathcal{M}) = \mathfrak{c}$ and empty interior
Suppose $\operatorname{cov}(\mathcal{M}) = \mathfrak{c}$, that is. union of less than continuum many meager subsets of $\Bbb R$is not $\Bbb R.$ I think the following is true but I can not see it...
View ArticleSolving for Eigenvalues of a Matrix with a Special Structure (Containing...
I am currently investigating the eigenvalue computation of a matrix with a special structure. Consider an $N$-dimensional matrix where all off-diagonal elements are $a$, diagonal elements are typically...
View ArticleIf $|A-B| < \epsilon$ for every $\epsilon > 0$ where $A, B$ are numbers, then...
Can you please check my proof of this statement using proof by contradiction?Claim:If $|A-B| < \epsilon$ for every $\epsilon > 0$ where $A, B$ are numbers, then $A = B$.Proof:Assume $A \neq...
View ArticleDerivative of $f(x)\cdot x$ where $f$ is a $\mathbb{R}^3$ function: how to...
Let $f:\mathbb{R}^3\to\mathbb{R}^3$. Could someone please help me to compute the the derivative of$$f(x)\cdot x?$$If $f$ would be a real valued function I think it should be the derivative of a...
View ArticleFinding the limit $\lim_{x \to 0}...
I was wondering how I could evaluate the expression $L$ as given below.$$\begin{equation}L=\lim_{x \to 0} \frac{1-\displaystyle\prod_{i=1}^n\cos^{1/i}{(ix)}}{1-\displaystyle \prod_{i=1}^n...
View ArticleIs the reverse inequality for Grönwall's inequality also correct?
I seems my question is different from this question. The version of Grönwall's inequality from wikipedia says if $u'(t) \leq \beta(t) u(t)$ then $u(t) \leq u(a) \exp(\int_{a}^t \beta(s) ds)$. Is it...
View ArticleExtensions preserving Lipschitz constant
Let $f:K \to \mathbb{R}$ be a $C^2$ function defined on the set $K=[0,1]^2$ (more generally, $K$ could be a convex and compact set in $\mathbb{R}^d$) such that $f$ and its derivatives are...
View ArticleIs this convex set open?
Let $X$ be a normed space over $\mathbb{R}$, and let $K \subset X$ be a convex subset with the property that, for every $u \in X, u \neq 0$ there exists $M(u) > 0$ such that$ \{ \lambda \in...
View ArticleOscillatory integral (stationary phase)
I am trying to integrate$$\int_{0}^{\infty}e^{-i n x^3}a(x)x^2dx$$Where $a(0)\neq 0$.Can I change variables $x^3\mapsto t$ to see that this an integral with no stationary points, and must got to $0$ in...
View ArticleProving that the second derivative of a convex function is nonnegative
My task is as follows:Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function, and let $f$'s second derivative be continuous. Let $f$ be convex with the following definition of convexity:...
View ArticleFolland question #1.2.5
Here is the question I am starring at:If $\mathcal{M}$ is the $\sigma$-algebra generated by $\epsilon,$ then $\mathcal{M}$ is the union of $\sigma$-algebras generated by $\mathcal{F}$ as $\mathcal{F}$...
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