Why can't $e^{x^2}$ be integrated [duplicate]
My teacher told me that not only do we have to use the erf function to approximate error, but that it is proved impossible to integrate in real analysis (at least not Riemann-integrable). Is there a...
View ArticleHow do I find the supremum and infimum of $\left\{x\in\mathbb{R} \setminus 0:...
$A=\{x \in \mathbb{R}: \frac{2x}{3}-\frac{x^2-3}{2x} + 0.5 < \frac{x}{6}, x \neq 0\}$I tried simplifying this inequality and got $x+3<0$, which is $x<-3$. But that means that either $x>0$...
View ArticleFind extreme points of $2x^3-12x+3y^2+6xy$ on restriction
Let's consider $f:\mathbb{R}^2\to\mathbb{R}$ with $f(x,y):=2x^3-12x+3y^2+6xy$. Find the global maximum and global minimum on $D:=\{(x,y)\in\mathbb{R}^2\mid x\geq 0,y\geq0,x+y\leq 1\}$.My...
View ArticleProving $x^3$ is Big-O of $x^2$ as $x \rightarrow 0$
EDIT: I have attempted the question again and posted an answer in a separate post if you scroll blow. Can you please check my NEW working?The definition I have is that $f(x)$ is Big-O of $g(x)$ as $x...
View ArticleClik here to view.
Rudin 4.22 Theorem
Could you help me understand why 1. f(H) = B and why 2. $\bar A$ $\cap$ B is empty andwhy 3. $\bar G$ $\cap$ H is empty?
View ArticleGeneralization of Fatou's Lemma Question in the proof of Royden [closed]
In this proof given by Royden in Real Analysis textbook, I have a hard time understanding why in case 1, the set $X_{\infty} \subseteq \cup_{n\in N}A_n$ and in case 2, the $\cup X_n=X$. Can someone...
View ArticleIf $\{f_n\}$ converges almost everywhere to $f$, and there exists $M>0$ such...
Prove that if $\{f_n\}$ converges almost everywhere to $f$, and there exists $M>0$ such that $|f_n(x)|\leq M$ almost everywhere, then $|f(x)|\leq M$ almost everywhere.How do I prove this? What kind...
View ArticleConvergence of probability measure with bounded second order momentum
We denote by ${\mathcal P}_1(\mathbb{R}^d)$ the set of Borel probability measures $\nu$ on $\mathbb{R}^d$ with a finite first order moment $M_1(\nu)$:$$M_1(\nu)= \int_{\mathbb{R}^d}...
View ArticleProve that there exists $ \xi \in [a, b] $ such that $ \int_a^b f g \, dx =...
Let $ f : [a, b] \to \mathbb{R} $ and $ g : [a, b] \to \mathbb{R} $ be functions such that $ f$ is continuous and $ g $ is monotonic, continuously differentiable, and non-negative on the interval $ [a,...
View ArticleProof that $0< e-\ln 2-2 < \frac 3 {100}$
You may have learnt the estimates $e\approx 2.7$ and $\ln 2 \approx 0.7$, thus yielding $e-\ln 2 \approx 2$. A calculator indicates more precisely that $e-\ln 2\approx 2.025$.Out of curiosity I'd like...
View ArticleProve inequality $ \frac{x^2}{(1 + x^2)^n}\ < \frac{1}{n}\ $ for any x [closed]
I'm trying to prove next inequality: $ \frac{x^2}{(1 + x^2)^n}\ < \frac{1}{n}\ $for $|x| < 1$ and any $n \in\ N$I've tried to use induction (can't make induction step) and taylors formula (I got...
View ArticleShowing a collection is a $\sigma$-algebra using pullback and pushforward
In my course, we have defined pullback and pushforward as follows. Let $f:X\to Y$ with $A\subseteq X$ and $B\subseteq Y$. Then the pullback $f^*$ of a collection of subsets $\{B_1,B_2,\cdots\}$ is...
View ArticleWhether it holds that ${\mathcal B}(\Omega)\bigcap \Omega_0={\mathcal...
Given a Probability space $(\Omega,{\mathcal B}(\Omega),P)$, where $\Omega$ is a Polish space and equipped with the distance$$d:\Omega\times \Omega \rightarrow [0,\infty).$$${\mathcal B}(\Omega)$ is...
View ArticleClik here to view.
Why the derivatives $f^{(n)}(x)$ of Flat functions grows so fast? (intuition...
Why the derivatives $f^{(n)}(x)$ of Flat functions grows so fast? (intuition behind)In this other question I did about Bump functions, other user told in an answer that these kind of functions "tends...
View ArticleProve that the sequence $\sqrt[n]{n!}$ diverges to infinity [duplicate]
Currently, I have that $\sqrt n \leq \sqrt[n]{n!}$ for all integers $n≥1$, and since the sequence in the lower bound $\lim\limits_{n\to \infty}\sqrt n=\infty $ blows up, so does the upper bound...
View ArticleContinuity of regular outer measures
Suppose $\mu^*$ is an outer measure induced from a premeasure on a ring generated by the semiring $\mathcal{H}$. Then one can show that $\mu^*$ is regular. How to prove that $\mu^*$ is continuous from...
View ArticlePointwise Convergence Definition Question
The definition of a sequence converging to a function on a set X is as follows:For all $\epsilon>0$ and for all $x\in X$, there exists a natural number $N$, such that $|f_n(x)-f(x)|<\epsilon$ for...
View ArticleExtension of sum of premeasures is sum of extensions of premeasures
Let $ \mu_1, \mu_2$ be two premeasures on a ring $\mathcal{R}$. Prove that$$(\mu_1 + \mu_2)^* = \mu_1^* + \mu_2^*$$ where $\mu^*$ is the Caratheodory extension of $\mu$, that is $$\mu^*(A) =...
View ArticleIntegrable funtion which is discontinuous on Cantor Set.
We know by Riemann Lebesgue theorem that any bounded funtion $f:[a,b]$$\to R$ is Riemann Integrable if the set of discontinuity of $f$ is a measure zero set.Now my question is : Is there any function...
View ArticleAre all continuous functions that send conics to conics in $\mathbb{R}^2$ a...
Motivation:Consider an arbitrary conic section in $\mathbb{R}^2$ given by$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$Now consider the map $$\phi: \mathbb{R}^2 \rightarrow \mathbb{R}^2,...
View Article