$f:[0,1] \to R$ is measurable but not Lebesgue integrable. Is $g(x,y) := f(x)...
Let's take $f:[0,1] \to \mathbb{R}$ as a measurable and not Lebesgue integrable function. Is that true that the function $g$ defined by the formula $g(x,y) := f(x) - f(y)$ is not Lebesgue integrable on...
View ArticleTo show elements of a set are bounded satisfying certain property.
Let S = { f : $\Bbb R\rightarrow\Bbb R$ | $\exists$$\epsilon$> 0 such that $\forall\delta > 0, |x - y| <\delta\implies |f(x) - f(y) |<\epsilon$ }. Then(a) S = { f :...
View ArticleQuestion on the Measurability of $\mathbb{R}^d$-Valued and Complex-Valued...
I am self-studying measure theory using Measure Theory by Donald Cohn. I got confused by the following two remarks made in this book:Example 2.6.5$\quad$ Let $(X,\mathscr{A})$ be a measurable space,...
View ArticleQuestion on Complex Integral with Polar Form
Let $f$ be a complex-valued integrable function. Write the complex number $\int fd\mu$ in its polar form, letting $w$ be a complex number of absolute value 1 such that\begin{align}\int fd\mu =...
View ArticleStrict proof of infinitesimal equivalency between $\sin{x}$ and $x$
When I was teaching infinitesimal equivalency between $\sin(x)$ and $x$ ($x\rightarrow0$) for Calculus, I realized that it was not very easy to have a pure elementary proof for it without using the...
View Articlehayman admissibility of e^z
Here is the definition of hayman admissibility according to generatingfunctionology (188-189, pdf online):Let $f(z) = \sum_{n=0}^{\infty} a_n z^n$ be regular in $|z| < R$, where $0 < R \leq...
View ArticleIs $U_a=d^{-1}([0,a])= \{ (m,n) \in M \times M: d(m,n) \leq a \}$ open set in...
Is $U_a=d^{-1}([0,a])= \{(m,n) \in M \times M: d(m,n) \leq a\}$ open set in a metric space $M \times M$ for positive real $a$?My attempt: I think it is not an open set. Because let $M=\mathbb{R}$ and...
View ArticleSuppose $x_n < 0$ for all $n$. If $\lim(x_n) = 0$, prove that $\lim(1/x_n) =...
How do you go by proving this one . any hint ?I have no idea where to start with this one
View ArticleShow that f is periodic if $f(x+a)+f(x+b)=\frac{f(2x)}{2}$?
Suppose $a$ and $b$ are distinct real numbers and $f$ is a continuous real function such that $\frac{f(x)}{x^2}$ goes to 0 when $x$ goes to infinity or minus infinity. Suppose that$...
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What is the reason for these strange oscillations? Issue with Desmos?
Take a partition of $\Bbb R^2_{\gt 0}$ by the union of functions indexed by real $t\ge 0$$$\mathcal F:=\bigg \lbrace \mathcal M[\chi_t(x)]\cup \mathcal M\bigg[\frac{1}{1-\chi_t(x)}\bigg] \bigg...
View ArticleDuals of Hilbert Subspace
So I am confused about something very basic. I'm going to outline my confusion, and would love if someone could point out when I'm saying something wrong.Let $H$ be a Hilbert space. It's dual $H^*$ can...
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An infinite product function
Before I start, must be said that I am not a math wizard, nor a math student. I just love nudging around with math, and I came across a random function I thought in my head (I do not take credit for...
View ArticleIf $f$ is in $L^2$, is its average in $L^2$? [duplicate]
Suppose $f$ is a continuous function on $[1,\infty)$ such that$$\int_1^{\infty} f(x)^2 dx <\infty$$Is it true that$$\int_{1}^{\infty} \frac{1}{r^2} \left( \int_{1}^r f(x) dx\right)^2 dr <...
View ArticleApproximation by smooth functions - do the derivatives converge locally...
Say I have a function $f:\mathbb{R} \to \mathbb{R}$ which is continously differentiable and has a bounded derivative. Then I know I can approximate $f$ with smooth functions $\phi_n$ by mollifications...
View ArticleThe isolated critical point!
I am confuse on the deffinition for a critical point which is called isolated. Please tell me what the isolated critical point?
View ArticleMinimum of $\frac{x^3}{x-6}$ for $x>6$ without using derivative?
Find the minimum of $y=f(x)=\dfrac{x^3}{x-6}$ for $x>6$.I can solve the question using derivatives but I have no any idea how to do it without them. Using derivatives, we find $x=9$ and...
View ArticlePreserved linear independence of vectors
I have been considering the following problem in Multivariable Calculus:Consider linearly independent vectors $a_1,$$a_2 ... a_k$ in $\mathbb{R}^n$. Show that there exists $\delta >0$ such that if...
View ArticleProving the Fundamental Theorem of Calculus for Step Functions
I'm hoping someone could look over my proof attempt of the following claim.The StatementFor the step function $\phi$ on the compatible partition $P=\{p_0,...p_k\}$. Then we say that the function...
View ArticleShowing $\big\{(x,y)\in\mathbb{R}^n$ $\times$ $\mathbb{R}$ $\mid $ $\min (...
In a course on Multivariable Calculus, I came across the following problem, and am looking for some guidance on how to approach it.QuestionConsider continuous functions $f,g$ in $\mathbb{R}^n$ and...
View ArticleWhat are these infinite sums of powers of integers, $n^p$, multiplying a...
What are explicit elementary functions of real $x$, for $0 < x < 1$, if they exist, for $p=1$ and $p=3$ of$$\sum_{n=1}^\infty n^p [J_n(nx)]^2$$$$\sum_{n=1}^\infty...
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