Is boundary of a discrete set in R is countable? [duplicate]
I have found plenty examples where boundaries of discrete set in R^2 is uncountable, but I couldn't find any such examples in R. Can u help me?
View ArticleExistence and uniqueness criterion for a specific first order ODE and...
Not being an experienced person in differential equations, I would like to clarify some issues for the following type of equations.Let we have the following equation$$y' = \frac{g(y)}{f(x)},$$where...
View ArticleProve that $\ln(x)$ is continuous at $1$ and at any positive real number $a$
Definition: We say that a function $f$ is continuous at a provided that for any $\epsilon > 0$, there exists a $\delta > 0$ such that if $|x−a| < \delta$ then $|f(x)−f(a)| < \epsilon$.(a)...
View ArticleHow can we find a limit point of a set? [closed]
How can we find a limit point of a set ? Is there any formal method or we have to observe the set , how it behaves?Like here , we have an example: $S=\{{1}/{2^m}+1/2{^n}: m,n \in \mathbb{Z} \}$. We...
View ArticleConvergence of Conditional Probability a.s.
I am looking for similar results for point-wise convergence of quantile function, but with a different setting.Suppose $\{B_n\}$ is a series of random variables, $B_n>0$ for each $n$. Let $Z$ be a...
View ArticleProve by epsilon delta method that $f(x) = \log(x)$ is continuous at $x=2$...
I am confused, about how to proceed with this question, I got this far only and got stuck :For any $\varepsilon>0$, there exist $δ>0$ such that $$|\log(x) - \log(2)|<\varepsilon...
View ArticleI need to prove the continuity of $f(x)=\log x$ using a $\epsilon$-$\delta$...
I need to prove the continuity of $f(x)=\log x$ using a $\epsilon$-$\delta$ proofThese is what I have so far but am not sure how to continue$|\log x-\log a| < \epsilon$$\log a- \epsilon < \log x...
View ArticleIs it possible to "sort" a continuous function?
I was motivated for this question while seeking for a new sorting algorithm.Suppose a continuous function $f : [a, b] \to \mathbb{R}$ is given. I wanted to define the sorted version $g$ of $f$, which...
View ArticleClik here to view.
Proving that a "convolution" series converges
I am attempting to prove the following questionIn my proof, it gets very technical, and I am unsure if I have manipulated inequalities correctly. Also, I am unsure if this proof is readable, as I have...
View ArticleClik here to view.
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 ArticleEvaluate $\int^{\pi}_0\frac{x\sin(x)}{1+\cos^2(x)}dx$ [duplicate]
I have the following task:On an interval $[0,a]$ one can use the substitution $y=a-x$ to try and exploit symmetry about the midpoint $a/2$1) Evaluate $\int^{\pi}_0\frac{x\sin(x)}{1+\cos^2(x)}dx$Where...
View ArticleIs there a generalization of second Mean Value Theorem of Integrals for...
The Second Mean value theorem of integrals states that:If $f,g:[a,b]\to\mathbb{R}$ are integrable functions such that $g$ is monotone, then there exists $x_0\in [a,b]$ such that $\displaystyle\int_a^b...
View ArticleStrictly increasing family of sets $\mathcal{E_j}$ which consists of...
In Folland's Real Analysis, Section 1.6, it saysOur characterization of the $\sigma$-algebra$\mathcal{M}(\mathcal{E})$ generated by a family $\mathcal{E} \subset \mathcal{P}(X)$ is nonconstructive, and...
View ArticleLocally Lipschitz with respect to a variable uniformly to another implies...
Let $$f:A\subset{\mathbb{R}^{n+1}}\to{\mathbb{R}^{n}}$$$$(t,\mathbf{y})\to{f(t,\mathbf{y})}$$ with $A$ open, $t\in{\mathbb{R}}$ and $\mathbf{y}\in{\mathbb{R}^{n}}$.The function $f$ is said to be...
View ArticleHow to prove that $\ell_{\infty}$ has no Schauder basis by contradiction?
I have a question about:schauder basis for $\ell_\infty$I know that $\ell_\infty$ is not separable, therefore has no Schauder basis.However I cannot understand why the set $\{e_1, e_2, e_3, \dotsc \}$...
View ArticleIf $|y_m| \to 0$ then there exists $k_m$ such that $k_m |y_m| \to t$
Lemma 3.43 Suppose $B_m$ are elements of $G$ and that $B_m \to I$. Let $Y_m = \log B_m$, which is defined for all sufficiently large $m$. Suppose that $Y_m$ is nonzero for all $m$ and that $Y_m/\|Y_m\|...
View ArticleA product of bounded and null sequences is a null sequence.
Assume $\{x_k\}$ and $\{y_k\}$ are two sequences in $\mathbb{R}^n$ such that the $\lim_{k \rightarrow \infty}x_k = 0$ and $\{y_k\}$ is bounded. prove $\lim_{k \rightarrow \infty} (x_k \centerdot y_k) =...
View ArticleNecessary condition for Gaussian KDE function to be nonnegative
Let $x_1, x_2, \dots x_n$ be fixed real numbers. Consider real numbers $v_1, v_2, \dots v_n$ such that $ v_1 + v_2 +\cdots + v_n > 0$. What condition do the points $v_1, v_2, \dots v_n$ need to...
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 ArticleIntegral $\int_0^1 \log \frac{1+ax}{1-ax}\frac{dx}{x\sqrt{1-x^2}}=\pi\arcsin a$
Hi I am trying to solve this integral $$I:=\int_0^1 \log\left(\frac{1+ax}{1-ax}\right)\,\frac{{\rm d}x}{x\sqrt{1-x^2}}=\pi\arcsin\left(a\right),\qquad\left\vert a\right\vert \leq 1.$$It gives beautiful...
View Article