Set of points where a continuous function is Holder is Borel.
A continuous function $f:\mathbb{R}\to\mathbb{R}$ is Holder at $x$ if there exists $C,\epsilon > 0$ and $\alpha\in(0,1]$ satisfying$$|f(x)- f(y)|\leq C|x-y|^\alpha,$$ whenever $|x-y| <...
View ArticleWhich is the correct approach to find if the sequence diverges? [closed]
So according to sources such as MathWorld and Wikipedia, as well as various Real Analysis textbooks, a sequence is defined as divergent if it is not convergent.But here's an interesting problem...Prove...
View Article$\lim_{n\to\infty} \sum_{r=1}^n \frac{n}{n+r} - n\log 2$
Consider the following two sequences of reals $$a_n=\frac{1}{n+1}+\cdots +\frac{1}{2n}$$ and $$b_n=\frac{1}{n}$$ How would one prove that $$\lim_{n\to\infty }\frac {a_n-\log 2}{b_n}$$ is finite and...
View ArticleOn the solutions of $\frac{\ln(x)}{x}=a$
Context: Given$$ f(x) = \frac {\ln(x)} {x} \; $$For $a\in\mathbb R$, if the equation $f(x) = a$ has two solutions points $x_{1}$ and $x_{2}$, i.e.$$f(x_{1}) =f(x_{2}) = a$$Calculate the range of...
View ArticleHow to find $\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$ and...
How to calculate$$\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$$and$$\sum_{n=1}^\infty\frac{(-1)^nH_{2n}^{(2)}}{n^2}$$by means of real methods?This question was suggested by Cornel the author of the book,...
View Articleprove a function is not Lebesgue integrable
Let $ f$ be a function defined on the interval $[a, b]$, and thereexist constants $ M > 0 $ and $ \alpha \geq 1 $ such that$$ |f(x)| \geq \dfrac{M}{|x - x_0|^\alpha} $$for some $ a < x_0 < b...
View ArticleAxiom of completeness counterexample for $A\subseteq \mathbb{Q}$
In Abbot's "Understanding Analysis," the Axiom of Completeness is stated as "every nonempty set of real numbers that is bounded above has a least upper bound." He then gives $S = \{r \in \mathbb{Q}|...
View ArticleDoes an asymptotically stationary curve in R^n with vanishing acceleration...
For twice continuously differentiable curves $\vec{x}(t) \in \mathbb{R}^n$, does$$\left( \lim_{t \to \infty} \vec{x}(t) = \vec{L} \quad \text{and} \quad \lim_{t \to \infty} \vec{x}''(t) = \vec{0}...
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Please provide an example of a groupoid
In the paper titled, 'Tannaka–Krein duality for compact groupoids I, Representation theory', the author proves the Peter Weyl theorem on compact groupoids. In the statement, he gives the hypothesis...
View ArticleA nonexpansive mapping and a pair of sequences
Let $a<b$ be real numbers and let $f:[a, b]\to [a, b]$ be a function such that $|f(x)-f(y)|\le |x-y|$ for all $x, y\in [a, b]$. Define the sequences $(x_n)_{n\ge 0}$ and $(y_n)_{n\ge 0}$ by...
View ArticleBig-Oh $\sin$ Proof
I want to prove that $\sin(x)=O(1)$ as $x \rightarrow \infty$I know there are different methods of solving. Are all these appropriate?Method $1$:I want to prove that $|\sin(x)| \leq K|1|$ whenever...
View ArticleFor which sets of functions can we establish $f$-mean inequality?
For a finite set $X \in \mathbb{R}$ and a bijective increasing function $f:\mathbb{R} \to \mathbb{R}$, let's define the $f$-mean of $X$ as the following.$$\mu_f(X) \equiv f^{-1} \left( \frac{1}{|X|}...
View ArticleClarification about continuity of cost function
I've come across the statement that the following function, referred to as Coulomb cost in optimal transport, is lower semicontinuous: $c : \mathbb{R}^{Nd} \rightarrow \mathbb{R}, c(x_1,.., x_N) =...
View ArticleFind a sequence of real-valued nonnegative functions fn on [0,1] such that...
Context: I am looking for a sequence of functions $(f_n)$ on $[0,1]$ such that $\lim \sup_{n\to\infty}f_n(x)=\infty$ for all $x\in[0,1]$ and $$\lim_{n\to\infty}\int_0^1f_n(x)\text{d}x=0$$Attempt: I'm...
View ArticleShowing that a recurrently defined sequence is convergent.
This is a text book question.Show that the sequence $\{u_n\}$ is convergent where $0<u_1<u_2$ and $u_{n+2}=\frac{2u_{n+1}+u_n}{3}$.I am aware that it's possible to find the formula of this...
View ArticleIs the multivariate function continuous?
Consider the following set: $x_1 \geqslant0, \cdots, x_N \geqslant 0, x_1+\cdots+x_N \leqslant 1$, denoted by $S$.Define a function $S \rightarrow \mathbb{R}$ as follows:For all the points that are not...
View ArticleCalculus criterion of strict convexity?
Suppose function $f\left(x\right)$ is defined on a closed interval $\left[a, b\right]$. Is "$f\left(x\right)$ is twice differentiable on $\left(a,b\right)$ and $f''\left(x\right)>0$ on $\left(a,...
View ArticleIf $f'(x) = O(1)$, then $f(x) = O(x)$
CONTEXTWhile self-studying Real Analysis I came across the following assertion, and tried to prove it.PROBLEMLet $f(x)$ be a differentiable function, defined in some right (left) neighborhood of $a$...
View Article$L^2$ property of samples
Assume a continuous, differentiable function $x(t)$ which is uniformly bounded and $L^2$ i.e.$\int_0^\infty{x^2(s)ds}<\infty$ and its derivative is also uniformly bounded. Do we also have that...
View ArticleShowing that $\mu(\phi^{-1}(V)) = \lambda (V).$
In a process of showing that the operator $T$ we gave here Showing that a set has measure zero? is ergodic, I want to prove that $\mu(\phi^{-1}(V)) = \lambda (V).$ where I have the following givens to...
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