Using Lebesgue's dominated convergence theorem to show a function is continuous.
I have a function $U(t)=\int_\mathbb{R} u(x) \cos(xt)dx$ and I am trying to use Lebesgue's dominated convergence theorem to show $U(t)$ is continuous for all $t \in \mathbb{R}$This is the proof. Notes...
View ArticleA domain in $\mathbb{R}^n$ with $C^2$-boundary satisfies an "outer spherical...
Let $\Omega\subseteq\mathbb{R}^n$ be a domain and $\partial\Omega\in C^2$, i.e.$\Omega=\overline{\Omega}^\circ$For all $x_0\in\partial\Omega$, there exists a neighbourhood $U\subseteq\mathbb{R}^n$ of...
View ArticleLimit of outer measures
Let $\mu^*$ be an outer measure on $X$. Let $\{A_n\}_{n=1}^{\infty}$ be a sequence in $X$ such that $A_1 \subset A_2 \subset ...$. How to prove that$$\lim_{n \to \infty} \mu^*(A_n) = \mu^*(A)$$where $A...
View ArticleIf $f'$ tends to a positive limit as $x$ approaches infinity, then $f$...
Some time ago, I asked this here. A restricted form of the second question could be this:If $f$ is a function with continuous first derivative in $\mathbb{R}$ and such that $$\lim_{x\to \infty} f'(x)...
View ArticleHow to prove that a product with constraints is greater than or equal to 1/2?...
I am working with the following expression:$$ \prod_{j=1}^{i-1} \left( 1 - \frac{1}{2} x_j \right) \geq \frac{1}{2}, $$where the variables $x_j$ satisfy the following conditions:$x_j \in [0, 1]$ for...
View ArticleSupremum of a function with power
Is it true that for a function $f(x): \mathbb{R} \to [0,\infty)$ it holds$$\sup[f(x)^a] \leq [\sup(f(x))]^a $$for $a \in (-\infty,+\infty)$?
View ArticleEvaluation of limit of integrals where calculus tricks don't work
Suppose that we want to evaluate $\displaystyle \lim_{n \rightarrow \infty} \int_0^1 \frac{n x^n}{1+x^3} \, dx$. Clearly the standard limit theorems for integration (Lebesgue dominated convergence...
View ArticleJacobi value from stereographic projection
Let $n\geq 3$. The stereographic projection from $\mathbb{S}^n \backslash\{N\}$ to $\mathbb{R}^n$ is the inverse of$$F: \mathbb{R}^n \rightarrow \mathbb{S}^n \backslash\{N\}, \quad x...
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$y\ln y - y = x\ln x - x$ ; How do I write $y$ explicitly as a function of $x$?
I used Desmos and found that $y\ln y - y = x\ln x - x$ can be splited into two functions.It looks clear that one of them is $y=x$.However, I am having trouble finding out the second one.How do I derive...
View ArticleProof of continuous image of compact set is compact
Theorem: Let $K\subset \mathbb{R}^n$ be compact and $f:K\to \mathbb{R}^m$ continuous. It holds that $f(K)$ is compact.I know the proof is contained in Spivak's Calculus on manifolds, however, I...
View ArticleShowing cos$(x^2)$ is analytic
In the textbook that I am using I am given the following definition and theorem:Definition: A function $f$ whose Taylor series converges to $f$ in a neighborhood of $c$ is said to be analytic at...
View ArticleN-Functions (Nice Young functions)
A mapping $\Phi:[0,\infty)\to[0,\infty)$ is termed an N-function (nice Young function) if(i) $\Phi$ is continuous on $[0,\infty)$;(ii) $\Phi$ is convex on $[0,\infty)$;(iii) $\lim\limits_{t \rightarrow...
View ArticleGeneral Gibb's Phenomenon Proof
I understand the easy proof of Gibb's Phenomenon where you first prove it for the (periodically extended) square wave $f(t) = \begin{cases} c & 0<t<1 \\ -c & -1 < t < 0 \end{cases}$...
View ArticleBartle Elements of Integration Exercise 7W
I have solved problem $7W$ in a particular case when $\varphi$ is uniformly continuous and I wonder if there are any counterexamples when uniform continuity isn't satisfied.7.W. If $f_n$ converges to...
View ArticleKernel feature and derivative of kernel feature linearly independent?
Suppose we have a strictly positive definite symmetric kernel $k$ on an open set $\Omega\subset\mathbb R$. By "strictly" I mean that all kernel matrices $(k(x_i,x_j))_{i,j}$ with distinct $x_i$ are...
View ArticleMinimum over a ball of a continuous function is continuous
I want to show that if $f:\mathbb{R}^n\to\mathbb{R}$ is continuous, then the function$$f_t(x)=\min_{\|x-y\|\leq t}f(y)$$is continuous for any $t>0$. I am having a hard time since the minimum point...
View ArticleExpressing trigonometric functions of infinitesimal arguments as algebraic...
I have recently been working with, and reading a bit about, infinitesimals and hyperreals and am currently trying to figure out how the trigonometric functions for infinitesimal inputs should behave...
View ArticleOrder Type of Totally Ordered Abelian Groups
As a preface, I don't know much group theory, so this might be a trivial question. I looked at some other questions on stack exchange, but I didn't notice a result which answers my question...
View ArticleIs exponentiation of isometries uniformly continuous in the uniform topology?
Let $(X,d)$ be a (bounded, complete, even polish) metric space, and let $G$ be the group of isometries of $(X,d)$ with the uniform metric $d_u$ defined by $d_u(f,g):=\sup_{x\in X}d(f(x),g(x))$ for all...
View ArticleHow to evaluate $\lim\limits_{x \to 0 }\frac{(\ln(1+ x)/x-1)}{x}$ without...
I want to solve this limit: $\displaystyle \lim\limits_{x \to 0 }\frac{(\frac{\ln(1+ x)}{x}-1)}{x}$This can be easily done with Taylor series (Since $\ln(1+x) = \sum_{m=1}^ \infty \frac{(-1)^{m+1}...
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