Proof of $f(\overline{S}) = \overline{f(S)}$
I know that $f(\overline{S}) = \overline{f(S)}$ if and only if $f: X \rightarrow Y$ is an homeomorphism, you can find a proof here.I am trying to establish a proof of the $\impliedby$ part, but I don't...
View ArticleLet $f\colon (a,b)\to \mathbb{R}$ be nondecreasing and continuous. If...
I need help to understand the proof below of the following theorem.Let $f\colon (a,b)\to \mathbb{R}$ be an arbitrary function. If $E=\{x\in (a,b)\mid f'(x)\text{ exists and }f'(x)=0\}$, then...
View ArticleA special identity relating poisson integral for L^2 boundary condition
Source: Marshall's Harmonic Measure Chapter 2 Problem 8I have been working on this problem the entire day, and I could not figure out a way to crack it.Question: Let $u(z)$ be the Poisson integral of...
View ArticleNatural number between two real numbers $y,x$ such that $y-x > 1$
Prove that for $x,y \in \mathbb{R}$ such that $y-x>1$ there is a natural number $n$ such that $x < n < y$.Consider the following set:$$S := \{n \in \mathbb{N} | x < n < y\}$$Suppose $S =...
View ArticleHow to understand uniform integrability?
From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?
View ArticleRudin's Principles of Mathematical Analysis, Theorem $2.41$.
Let $E$ be a subset of $R^{k}$, with the Euclidean metric on it. Then $1.$ implies $2.$, where$1.$ Every infinite subset of $E$ has a limit point in $E$$2.$$E$ is closed.Proof Suppose by contradiction...
View ArticleMass conservation for a transport equation in mild form
Let us consider the following partial differential equation for $f = f(x,v,t),$ with $x \in \mathbb{T}^1, v \in \mathbb{R}$ and $t \in [0, \tau]$:$$ \partial_t f(x,v,t) + v \cdot \partial_x f (x,v,t) =...
View ArticleDefinitions of "non-degenerative closed bounded intervals"
What is the definition of a non-degenerative closed-bounded interval? An example would also be extremely helpful.
View ArticleBump function with integral $1$ and value $1$ at zero
How can i contruct a smooth bump function $F$ on $\Bbb{R}^n$ such that $F(0)=1$ and with integral $1$?I have tried to manipulate the function $f(x)=e^{-\frac{1}{x^2}}$ if $x>0$ and $f(x)=0$ if $x...
View ArticleSignificance of Convergence Theorems in Analysis
We use theorems like Monotone Convergence, Dominated Convergence, Bounded Convergence theorems to show that when a sequence of measurable functions $f_n$ in a measure space $X$ converges to some...
View ArticleProblem when proving the series $1+\frac 13-\frac 12+\frac 15+\frac17-\frac...
Prove that the series $$1+\frac 13-\frac 12+\frac 15+\frac17-\frac 14 +\frac 19+\frac 1{11}-\frac 16+\cdots$$ converges to $\frac 32\log 2.$I tried solving the problem as follows:The series given is...
View ArticleSolving for an Exercise from Zorich's Mathematical Analysis I Chapter 5 [closed]
Let $f\in C^{(n)}((-1,1))$ and $\sup_{-1<x<1}|f(x)|\le 1.$ Let $m_k(I)=\inf_{x\in I}|f^{(k)}(x)|$ where $I$ is an interval contained in $(-1,1)$. Show that:a) if $I$ is partitioned into three...
View ArticleLocal property of an integration inequality to global result
Here is the question.Let $f$, $g$ be locally integrable functions on $\mathbb{R}^n$ such that$$\inf_{a \in \mathbb{R}} \int_{B} |f(x) - a|\ dx \leqslant \int_B|g(x)|\ dx$$for all balls $B$ in...
View ArticleProof regarding partial derivatives and differentiability
I'm having a trouble in understanding the proof of Theorem 8 from Pugh's Real Mathematical Analysis 2nd Edition (p. 284).On the 5th line from the bottom where use has been made of the chain rule and...
View ArticleDoes there exist a positive sequence with these two properties?
Let $\{x\}$ denote the fractional part of $x$. Does there exist a sequence with all positive terms $(a_n)_{n\geq 1}$ such that $$\lim_{n\to\infty} \{(-1)^n a_n\}=1\ \ \ \text{and}\ \ \...
View ArticleIs there an intuitive explanation for the equality of mixed partials?
The fact that the mixed second order partial derivatives of a $C^2$ smooth scalar valued function are equal seems, to me, quite surprising. For example, if you interpret $\frac{\partial ^2f}{\partial y...
View ArticleIf $x$ and $y$ are arbitrary real numbers with $x < y$, prove that there...
This is a question from Apostol Calculus section I.3.12 q1If $x$ and $y$ are arbitrary real numbers with $x < y$, prove that there exists at least one real satisfying $x < z < y$I've seen a...
View ArticleGiven a metric $d$, is continuity of $f$ absolutely essential for the...
Let $d$ be a metric on a nonempty set $X$, and let $f \colon [0, +\infty) \longrightarrow [0, +\infty)$ be a function such that (i) $f(0) = 0$, (ii) $f(r+s)\leq f(r) + f(s)$ for all $r, s \in [0,...
View ArticleBaby Rudin theorem 2.41
I am reading Baby Rudin chapter 2 and came across some question on Theorem 2.41.When Rudin tries to prove that every infinite subset of $E$ has a limit points in $E$ implies $E$ is closed, he first...
View ArticleTrouble Understanding Baby Rudin $2.38$
Theorem $2.38$ states the following:If $\{I_n\}$ is a sequence of intervals in $R^1$, such that $I_n$ contains $I_{n+1}$$(n=1,2,3,…),$ then the infinite intersection of the sets $\{I_n\}$ is not...
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