Identity with Dirac delta function: $\delta (x^2-a^2) =...
How can I show that $\delta (x^2-a^2) = \frac{1}{2|a|}[\delta(x-a)+\delta(x+a)]$? I'm suppose to integrate it by a differentiable function and integrate, but I can't figure this one out.
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Amann/Escher, Analysis I, Exercise I.10.16: nest of intervals
I am doing Exercise I.10.16 from textbook Analysis I by Amann/Escher.Could you please verify if my attempt contains logical gaps/errors?My attempt:Let $I_n := [a_n, b_n]$ for all $n \in \mathbb...
View ArticleAdvice and supplementary sources (videos, books) to tackle rudin's real and...
I just started grad school and I have a mandatory class in Calculus and Probability.Professor's plan is to use baby rudin (Principles of Mathematical Analysis) and papa rudin (Real and Complex...
View Article$f$ is a real function and it is $\alpha$-Holder continuous with $\alpha>1$....
Maybe this is a well know result, however, I could not find it. Before stating it, let me write here a well know result (at least for me)Assume that $\Omega\subset\mathbb{R}^N$ is a open domain and...
View ArticleProduct vs Sum of two Heaviside step functions
Let $\theta(x)$ be the Heaviside step function defined as$$\theta(x) := \begin{cases} 1 & x \geq 0\\0 & x \lt 0\end{cases}$$Then is $\theta(x)\theta(1-x)$ the same as $\theta(x)-\theta(x-1)$,...
View ArticleProve that $g(x)=0$ for all $x$ in $\mathbb{R}$
Suppose that the function $g:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and that $g(x)=0$ if $x$ is rational. Prove that $g(x)=0$ for all $x$ in $\mathbb{R}$.Proof:Let $\{x_n\}$ be a sequence of...
View ArticleWhy is "Cauchy's criterion without an absolute value" not the standard form...
Cauchy's criterion (for sequences) says that a sequence $(a_n)$ converges if and only if for every $\epsilon >0$ there exists a natural number $N$ such that for all natural numbers $m,n>N$ we...
View Article$C_0(\mathbf{R})\otimes C_0(\mathbf{R})$ is dense in $C_0(\mathbf{R}^2)$ with...
$\textbf{Problem:}$ Let $\mathfrak{X}:=\{u\otimes v:\;u,v\in C_0(\mathbf{R})\}$, where $u\otimes v$ is given by $$(u\otimes v)(x,y)=u(x)v(y),\;\forall x,y\in\mathbf{R}.$$Prove that the linear subspace...
View Article$\epsilon$-$\delta$ Verification of a Lipschitz Function
A function $f:D\rightarrow \mathbb{R}$ is said to be a Lipschitz function provided that there is a nonnegative number $C$ such that $|f(u)-f(v)|\le C|u-v|$ for all $u,v\in D$.Show that a Lipschitz...
View ArticleStein's Real Analysis Chapter 1 Exercise 34
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be any two Cantor sets (constructed in Exercise 3.)Show that there exists a function $F:[0,1] \to [0,1]$ with the following properties:1. $F$ is continuous and...
View ArticleStuck on a step in a proof in Friedman's Foundations of Modern Analysis
I'm reading Avner Friedman's Foundations of Modern Analysis and I am getting stuck on a step in the proof of the following theorem:Theorem 2.4.3 If a sequence $\{f_n\}$ of a.e. real-valued, measurable...
View ArticlePartition of an interval of $\mathbb{R}$
A partition of an inteval $[a,b]$ of $\mathbb{R}$ is generally defined as a finite sequence of the form:$$a = x_0 < x_1 < x_2 < \dots < x_n = b$$Then, $[a,b]$ is seen as the following union...
View Article$|f(x)|\leq\dfrac{M}{1+|x|^a}\quad (a>1)\quad\text{almost everywhere}$
If $f$ is measurable on $\mathbb{R}$ and there exists a number $M>0$ such that$$|f(x)|\leq\dfrac{M}{1+|x|^a}\quad (a>1)\quad\text{almost everywhere},$$then $f$ is Lebesgue integrable.I was hinted...
View ArticleIs there a maximal extended measurable space that maintains measure for given...
The precise description of the title is that:-Given $(X,\mathcal M,\mu)$, we say a extended measurable space $\mathcal M'$ that maitains measure, which means $\mathcal M \subset \mathcal M'$ and...
View ArticleIs the set $\{\sqrt n - \lfloor\sqrt n\rfloor : n\in \mathbb Z^+\}$ dense in...
I was reviewing basic Analysis and thought of this question:Is the set $\{\sqrt n - \lfloor\sqrt n\rfloor : n\in \mathbb Z^+\}$ dense in $[0,1]$?Lest we have any ambiguity with notational stuff,...
View ArticleIf $E-E$ contains an interval centered at 0, is $E$ necessarily measurable?
It is a fairly standard problem in Real Analysis (see, for example, Excercise 31 in Ch1 of Folland) to prove that if $E \subset \mathbb{R}$ is a Lebesgue measurable set such that $m(E) > 0$, then...
View ArticleProve that there exists $x_n$ such that $0 \leq x_n \leq 1-\frac{1}{n}$ and...
Suppose that the function $f:[0,1] \to \mathbb{R}$ is continuous on $[0,1]$ and $f(0)=f(1)$. Prove that for each natural number $n$, there exists $x_n \in \mathbb{R}$ such that $0 \leq x_n \leq...
View ArticleA kind of converse of the inverse function theorem?
As I understand it, the inverse function theorem says:Let $h$ be a continuously differential map from an open subset $U\subset \mathbb{R}^n$ into $\mathbb{R}^n$. If the Jacobian $Jh$ has nonzero...
View ArticleIs $f(x) = \sum_{k=0}^\infty \sin(3^kx)$ continuous? Is it Riemann integrable?
This function is almost like a Weierstrass function, except for $a = 1$, which means that its continuity or integrability are not touched upon by standard proofs of the continuity of the Weierstrass...
View ArticleCan the derived set of a countable set be the Cantor set?
I was asked to construct a sequence on $\mathbb{R}$ such that the limit points of the sequence are the Cantor set, or simply to prove that is impossible.Intuitively, I try to construct a countable set...
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