Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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Continuity and differentiability of $x^a\sin ({1\over x}) $ at $0$

Consider the function $$ g_a (x) = \begin{cases} x^a\sin ({1\over x}) & x \neq 0 \\ 0 & x=0 \end{cases}$$ I am looking to determine for which $a$ the map $g_a$ is differentiable on ...
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54 views

$T_n \rightarrow T$ then we have $||T|| \le liminf(||T_n||)$

I know how to show that a cauchy sequence of linear continuous operators $T_n:X \rightarrow Y$ has a limit that is also such an operator(if Y is a Banach space), but I found this relation here too ...
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53 views

Proof about rational sequences

I'm stuck on part (iii). I wanted to use induction, but i'm having trouble proving the base case, which I took as n=3. Perhaps this was the wrong idea, but we'll see. The only thing we know ...
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1answer
33 views

Direct sum decomposition of $l_2$

Let $X=(V,E)$ be a finite graph and a linear operator $\nabla: l_2(V) \to l_2(E)$ given by the formula $(\nabla f)(x,y)=$ \begin{cases} f(x)-f(y) &d(x,y)=1\\ 0 &\text{otherwise} ...
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1answer
41 views

Small question about inductive proof about rational sequences

I am writing an inductive proof about this: the description is not terribly important so you don't have to read that. here's my question: let $P(n)$ be the statement that $x_n$ is a rational ...
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1answer
52 views

Analytical solution for nonlinear equation

Simple question: Does $\alpha = \frac{x}{\beta} - \left(\frac{x}{\gamma}\right) ^{1/\delta}$ have an analytical solution? ($\alpha,\beta,\gamma,\delta$ are constant) I'm working on big data arrays ...
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1answer
27 views

An unknown function

I run into a function: $1_{[-n, n]^r}$. I guess this function equals 1 whenever x falls into $[-n, n]^r$. Am I right? I met this function in an analysis paper which deals with measure and density of ...
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1answer
49 views

How can I show that $b_n \rightarrow a$? [duplicate]

Knowing that $a_n \rightarrow a$ and $b_n=\frac{a_1+a_2+...+a_n}{n}$. How can I show that $b_n \rightarrow a$?
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98 views

Textbook has wrong answer? - Metric spaces “topological properties” (probably trivial for the confident)

In the book there's a table and above it it reads "we have crossed out the wrong answer" meaning the remaining one is right. I dispute this, there are 4, I thought I got the first one right, but I ...
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3answers
63 views

Derivative of functions at a point

If the function $f:[0,1]\to \mathbb R$ defined by $f(x)=\dfrac{\sin x}{x}$ when $x\neq 0$ and $f(0)=0$, then is $f$ differentiable at $x=0?$ I think by using L'Hospital's rule $f$ is differentiable ...
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33 views

which solution is the right one?

Let $0<x_{1}<1$ and $x_{n+1}=1-\sqrt{1-x_{n}} $.Show that $(x_{n})$ is decreasing.Find the limit of $x_{n}$ and the limit of $\frac{x_{n+1}}{x_{n}}$. I showed that ...
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169 views

Show that $\sup(A \cup B)=\max\{\sup A,\sup B\}$

Let $A,B$ not empty,bounded subsets of $\mathbb{R}$.Show that $$\sup(A \cup B)= \max \{\sup A, \sup B \}.$$ That's what I have done so far: Let $x\in A \cup B \Rightarrow x \in A \text{ or } x\in B ...
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274 views

Property of slowly varying functions.

A function slowly varying at infinity $L:(0,\infty)\rightarrow (0.+\infty)$, has the property that for any $\delta>0$ if $x$ is large enough, then $L(x)\leq x^\delta$. Does this property implies ...
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63 views

Making $g(x)=x^2\operatorname{sgn}(x)$ continuous at $0$.

How would the function $g(x) = x^2 \operatorname{sgn} (x) $ be defined at $x= 0 $ so that it is continuous there? This makes no sense to me. It isn't, period. Why would it be? I can't re-define the ...
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1answer
25 views

Average value of a density function

I want to compute an average density, but only of the yellow object on the picture (1< x^2+y^2<9) between circles and y>=0 The density of the object is a function $$p(x,y)=\frac{y}{x^2+y^2}$$ ...
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144 views

Proof: $C(X×Y)=C(X)⊗C(Y)$

Where I can find the proof of the following theorem: Let $X$ and $Y$ be compact Hausdorff spaces, $C(X)$ and $C(Y)$ the space of continuous functions on $X$ and $Y$ respectively, then we have ...
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1answer
52 views

Show $\int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0$

let $\rho(x)=\sqrt{x}, \hspace{4mm} \forall x \in \mathbb{R}$ Show : $$ \int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0. $$ My attempt: \begin{align*} ...
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2answers
65 views

Power Series of $\sin x$, how to show positive.

I have been asked to show that $S(x)$, the power series for $\sin x$, is $>0$ for $0 < x \le \sqrt6$. I have altered the series into the form ...
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1answer
60 views

relation between topologies

I Came across this question recently. Could someone help me out with this. Let T1 be the smallest topology on $\mathbb{R}^2$ containing the sets $(a,b)\times (c, d)$ for all $a,b, c, d$ in ...
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1answer
202 views

Prove Borel's Lemma (Pugh's book #35) [duplicate]

Given any sequence whatsoever of real numbers (a_r), there is a smooth function $f: \mathbb{R} \to \mathbb{R}$ such that $f^{(r)}(0) = a_r$. Pugh's hint says to try $f=\sum \beta_k(x)a_kx^k/k!$, ...
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1answer
89 views

Applications of the theory of distributions outside of PDEs?

Are there any interesting, important or powerful mathematical applications to the Theory of Distributions besides those dealing with partial differential equations?
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1answer
90 views

Poisson Integral of a Lipschitz continuous function

I am reading a paper that makes reference to the following fact: Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous of some positive order $\alpha$. Let $H(x,y)$ be the extension of ...
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1answer
392 views

How to show that the set of all Lipschitz functions on a compact set X is dense in C(X)?

Im reading Chapter12 of Carothers' Real Analysis, 1ed. Here is a reading material of Lip(X) which denotes the set of all Lipschitz functions on a compact set X, How to show that the set of all ...
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1answer
54 views

Weak Convergence Proof: Tried on My Own

This is related to a question asked here: What a proof of weak convergence is supposed to look like I asked what a proof of weak convergence was "supposed to look like". Specifically, I asked that if ...
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760 views

What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers?

From Stephen Abbott's Understanding Analysis: Theorem: There is no rational number whose square is 2. Proof: Assume for contradiction, that there exist integers $p$ and $q$ ...
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1answer
35 views

Checking the solution of a first order pde

I need some help with this exercise. Given the following pde: $ \begin{cases} u_t + b(u)\cdot u_x=0\\[6pt] u(x, 0) = u_0(x) \end{cases} $ I have to check that its solution is $u(x, ...
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51 views

Proving uniform convergence of a sequence of functions

Let $f:[0,1]\to \mathbb{R}$ be continous for $n \geq 0$ and $0 \leq t \leq 1$. define $f_n(t) = f(t)t^n$. Prove that if $f(1)=0$ then $f_n \to 0$ uniformly on $[0,1]$. How do we prove it?
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48 views

What is the radius of convergence of the power series?

I have the following power series and I would like to figure out the radius of convergence: $$\sum_{n=1}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$$ I appreciate any help&explanation. Jacky
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145 views

Question about quantifier logic

This is my first post on the mathematics stack exchange so please bear with me.. I am new to quantifier logic and I just can't seem to wrap my head around it. I have been given four statements and I ...
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3answers
401 views

Can we call domain as inverse image of a function?

I was going through the definition of inverse image of a function http://www.northeastern.edu/suciu/U565/MATH4565-sp10-handout1.pdf, and I was wondering if inverse image of a function is the domain of ...
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37 views

How to determine the Radius of convergence?

I have the function $$ \sum_{n=1}^{\infty} \frac{(-3x)^n}{n^2}$$ I´m not realy sure where to begin and how to determine the radius of convergence. Could someone provide a nice explanation? THX
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94 views

Biholomorphic map between $D(0,1)\smallsetminus[0, 1)$ and the upper half plane

I want to find a biholomorphic map between $D(0,1)\setminus[0, 1)$ and the upper half plane. The problem is what to do with $0$. Any hint ?
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1answer
61 views

How do i prove this property of Riemann-integrable function?

Let $f$ be a Riemann-integrable function. That is, it's upper sum and lower sum are the same. Then, how do i prove that $\int_a^b f dx = \lim_{n\to\infty} \sum_{i=1}^n ...
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1answer
69 views

Being confused by a definition of Lattice

Im reading Chapter12 of Carothers' Real Analysis, 1ed talking about the Stone-Weierstrass theorem. Here is a definition of Lattice, See. every pair of elements has both a sup and inf in the lattice. ...
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1answer
53 views

Bijective function- for reals on set [0,0.1)

Find a bijective function from the reals to the set [0, 0.1) I think Cantor has something to do with it but I am unsure? Thoughts?
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154 views

Gauss Hermite quadrature on finite interval

I would like to approximate an Integral of the type $I = \int_{x_l}^{x_u} f(x) w(x) dx$ where $w(x) = \frac{1}{2\pi}e^{-\frac{1}{2}x^2}$ and $f(x)$ is only defined on the Intervall $D = [x_l, \, ...
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99 views

Remainder of Taylor series

The Taylor series of the function $$f(x) = \int_{1}^{\sqrt{x}} \ln(xy)+ y^{3x} dy + e^{2x}$$ at the point $x = 1$ is $$e^2 + (x-1)\left(2e^2+\frac{1}{2}\right) + ...
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1answer
124 views

Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0<\alpha<1$ an interval $I$ such that $m(A\cap I)/m(I)>\alpha$.

Let $A\subset\mathbb{R}$ a measurable where $0<m(A)<\infty$. Show that exists for each $0<\alpha<1$ an interval $I$ such that $$ \frac{m(A\cap I)}{m(I)}>\alpha. $$ MY ATTEMPT: ...
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1answer
136 views

Prove $y = x$ is continuous

For every $\epsilon > 0$ there exists a $\delta > 0$ such that $|x - c| < \delta$ implies $|f(x) - f(c)| < \epsilon$. Start with $|f(x) - f(c)| < \epsilon$ which gives $|x - c| < ...
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3answers
23 views

Limits Convergence and Absolute convergence

If $a_{n}$ is an arithmetical progression and $|\lambda|<1$ then how can we prove that $\lim_{n\to\infty}a_{n}\lambda^{n}=0$.
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53 views

Defining the Cantor Set

I'm having trouble understanding the construction of the Cantor Set as defined by wikipedia. In particular, we have that $$ C_0 = [0,1] $$ and then $$ C_1 = [0, \frac{1}{3}] \cup [\frac{2}{3}, 1] ...
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31 views

Convergence of $f_{n}(x)=\frac{1}{x^2+n^2}$ and $g_{n}(x)=\frac{2nx}{x^2+n^2}$ in sup norm

I need to show that (i) $f_{n}(x)=\frac{1}{x^2+n^2}$ converges to the zero function in sup norm, and (ii) $g_{n}(x)=\frac{2nx}{x^2+n^2}$ does not. Not sure if this is right but would appreciate ...
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1answer
111 views

Commuting Exponential Matrices

Let $x(t)=\exp(tA)\exp(tB)$ and $y(t)=\exp(t(A+B))$. Show that if $AB=BA$ then $x(t)$ and $y(t)$ satisfy the same initial value problem for ODEs and therefore must be equal. $A, B$ square matrices. ...
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104 views

The dimension of the real continuous functions as a vector space over $\mathbb{R}$ is not countable?

This question is out of curiosity. I first attempted a web crawl for this answer but was befuddled when Google didn't turn up the result after a couple of tries. If anyone has a reference, I'd be ...
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1answer
33 views

About convergence and uniform convergence of three series

I am not sure about the convergence of the third one and the uniform convergence of all three series
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49 views

Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.

Show that there exists a not decreasing function $f:(a,b)\rightarrow\mathbb{R}$ continuous on $(a,b)\setminus D$ and discontinuous on $D$ where $D$ is a countably infinite subset of $(a,b)$. This is ...
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1answer
45 views

Convergence almost everywhere and absolute value

Is it true that if $(f_n)$ converges almost everywhere to $f$ then $(\vert f_n \vert)$ converges also almost everywhere to $\vert f \vert$ ? Thanks.
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1answer
63 views

A condition implying $\phi $ holomorphic is linear

Let $\Omega$ be a bounded open subset of $\mathbb{C}$, and $\phi : \Omega \rightarrow \Omega$ holomorphic. Prove that if there exists a point $z_{0} \in \Omega$ such that $$\phi(z_{0}) = ...
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2answers
1k views

What is a limit point

Wikipedia seems to describe the topic with extreme complexity for me. In mathematics, a limit point of a set $S$ in a topological space $X$ is a point $x$ (which is in $X$, but not necessarily ...
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1answer
55 views

understanding pointwise convergence a.e. in measure theory

I'm going through a proof and the assumptions are that $\mu$ is a complete measure and that $f_n\rightarrow f$ $\mu$-a.e. One of the lines in the proof says If $f_n\rightarrow f$ $\mu$-a.e., then ...