Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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4answers
41 views

prove the continuous and open

Here is the question Suppose $E$ is a subset of $R$ and $f,g :E \rightarrow R$ are continuous on $E$. Show that $\{x \in E : f(x) > g(x)\} $ is open I'm confusing with this question. what ...
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1answer
40 views

$f(0)$ is integral over Fourier transform for Schwartz class

Let $S$ denote the Schwartz class. Assume without proof that for every $f,g\in S$, we also have $\hat{f},\hat{g}\in S$, and $\int_\mathbb{R}f(y)\hat{g}(y)dy=\int_\mathbb{R}\hat{f}(t)g(t)dt$. Show that ...
4
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1answer
74 views

Passing limit inside integral for functions in $L^1+L^2$ norm

Let $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, and let $f_k$ be functions in the Schwartz class such that $\|f-f_k\|_1+\|f-f_k\|_2\rightarrow 0$ as $k\rightarrow\infty$. Define ...
0
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1answer
66 views

Function is Measurable if and only if the restricted function is Measurable

Let $g:E\to \mathbb {R}$ be a function on a measurable set $E$ and let $\{E_n : n\in \mathbb{N}\}$ be a family of measurable subsets of $E$ such that $E=\cup_{n=1}^\infty E_n$. Prove that $f$ is ...
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1answer
51 views

Help understanding proof involving smooth functions of compact support

In the following proof I'm having trouble with two things: Why the initial test function $\eta$ should be chosen to have integral $=1$. How we know $\psi$ is smooth of compact support. I'm ...
3
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1answer
83 views

$L^2$ norm of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$f_r(\theta)=\sum_{n=1}^\infty r^na_ne^{in\theta}$$ is a harmonic ...
1
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1answer
93 views

Wirtinger's inequality in higher dimension

Wirtinger's inequality for one-dimensional functions states that if $f(x)$, $f'(x) = \frac{df(x)}{dx}$ $\in$ $\mathcal{L}^2(a,b)$ and either $f(a) = 0$ or $f(b) = 0$ then \begin{equation} \int_{a}^{b} ...
12
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2answers
188 views

Showing a set is countable or not.

Let $A=\{x\in\mathbb{R}:\forall n\in\mathbb{Z}^+,\lfloor x^n \rfloor \text{ is odd} \}$ where $\lfloor x \rfloor$ is the largest integer equal or less than $x$. Is $A$ countable?
2
votes
1answer
287 views

Use Fourier transform to calculate double integral of harmonic function

Let $$P_y(x)=\dfrac{1}{2\pi}\int_{-\infty}^\infty e^{-y|t|}e^{ixt}dt=\dfrac{1}{\pi}\dfrac{y}{x^2+y^2}.$$ Then $P_y(x)$ is harmonic in the upper half-plane $y>0$ and for $f\in L^1(\mathbb{R})$, ...
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1answer
38 views

Proving some two sequences is going to $0$

Let $\{a_{n}\}$ and $\{b_{n}\}$ be two sequences of a real numbers. If $\lim_{n \to \infty} a_{n} = 0$, then prove $$\lim_{n \to \infty} a_{n} \sin(b_{n}) = 0 $$ $|\lim_{n \to \infty} a_{n} ...
2
votes
1answer
45 views

Integral of $L^2$ function is continuous

For $f\in L^2(\mathbb{R})$, denote $$s_N(x)=\dfrac{1}{2\pi}\int_{-N}^N\hat{f}(t)e^{ixt}dt.$$ I'd like to prove that the integral converges, and that $s_N$ is continuous. Since $f\in ...
1
vote
1answer
54 views

existence of positive integer and converges to $L$

if for any given $0 < \epsilon < 1$ there exists a positive integer $N$ so that $|a_n - L| < 5\epsilon$ when $n^2 +1 > N $, then $\{a_n\}$ converges to $L$. is this true or fase? I do ...
0
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1answer
317 views

Details for proof of Poisson summation formula

In the proof of the Poisson summation formula, there is a detail which is not clear to me how to resolve. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Schwartz-class function. Let ...
0
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2answers
55 views

$\lim_{n\to \infty} \int^1_0 f_n(x)dx=0$ , is $f_n$ pointwise convergent?

Let $f_n(x)$ be a sequence of continuous non negative functions on $[0,1]$ such that $\lim_{n\to \infty} \int^1_0 f_n(x)dx=0$ . Then does $f_n$ necessarily converge pointwise? I think it is not ...
0
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1answer
27 views

Identity approximation for functions in $C_0(\mathbb{R})$

Let $K\in L^1(\mathbb{R})$ satisfy $\int_\mathbb{R} K(x)dx=1$, and denote $K_a(x)=\frac1aK(\frac{x}{a})$ for $a>0$. We get trivially that $\int_\mathbb{R}K_a(x)=1$. Let $x\in\mathbb{R}$ and $f\in ...
2
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0answers
27 views

$f(x)=x^{\frac{2}{3}}\ln x$ for $x>0$ , is it uniformly continuous [duplicate]

Let $f$ be the real valued function 0n $[0,\infty]$ defined by $f(x)=x^{\frac{2}{3}}\ln x$ for $x>0$ and $0$ for $x=0$. Is it uniform continuous on $[0,\infty)$? I think that if we break ...
0
votes
2answers
54 views

Closed subset of R^2

Show that the set A = {(x,y): $x^3$ $>=$ $y^5$} is closed as a subset of $R^2$. I defined a closed set as a set whose complement is open. So the complement of the above set is {(x,y): $x^3$ ...
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0answers
26 views

bounded * o = 0 , limited point prove

suppose E $\subset$ R and p is limited point of E. and f,g are real valued function on E if g is bounded on E and $lim_{x \to p} f(x) = 0 $ $$lim_{x\to p} f(x)g(x) = 0$$ I think I'm doing some part ...
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3answers
82 views

Proving that $\lim_{x \to \infty} f(x) = \frac{\sqrt[3]{x^3+2}+\sqrt[3]{8x^3+1}-3x}{2}=0$

I stumbled upon a limit some time ago today, which I've tried solving with no success: $$\lim_{x \to \infty} f(x) = \frac{\sqrt[3]{x^3+2}+\sqrt[3]{8x^3+1}-3x}{2}=0$$ Which presents an $\infty-\infty$ ...
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2answers
103 views

Question on Lebesgue measure

Say a set $E$ is $L$-measurable if for all bounded open intervals $(a,b)$ we have $b−a=m^*((a,b)\cap E)+ m^*(E\cap(a,b)c)$. How do we show that any $L$-measurable set is Lebesgue measurable? Could ...
2
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2answers
87 views

Tricky differentials problem involving continuous functions

Suppose $f$ is a continuous function on $[0, \infty )$, differentials on $(0, \infty)$, such that $f(0)=1$ and $f'(x)> \frac{1}{2\surd (x+1)} \forall x>0$. Show that $f(x)> \surd (x+1)$. ...
5
votes
2answers
128 views

How to determine the convergence of $\sum\limits_{n=1}^{\infty}\frac{n^{n^2}}{(n+x)^{n^2}}$

I've been studying for my analysis exams, and I've come across a series I haven't been able to solve. The question is just to determine for which real values of $x$ does the series converge. I've ...
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3answers
142 views

Uniform convergence of $\sum\limits_{n=1}^\infty \frac{(-1)^n}{ne^{nx}}$

Prove that $$ f(x) = \sum\limits_{n=1}^\infty \frac{(-1)^n}{ne^{nx}} = \sum\limits_{n=1}^\infty \frac{(-e^{-x})^n}{n}$$ is uniform convergent for $x \in [0,\infty)$. Attempt: At first, this looked ...
3
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0answers
44 views

Norms on $\mathbb{Q}$

So with respect to the metric $d(x,y)=|x-y|$ induced by the standard absolute value, the real numbers can be constructed as a completion of $\mathbb{Q}$. With respect to the metric $d_p(x,y)=|x-y|_p$ ...
1
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1answer
46 views

Convergence of $\int_{0}^{2}{k^{p}x^{k} \over 1 + x^{2k}}\,{\rm d}x\quad\mbox{with}\quad k = 1,2,3,\ldots\quad\mbox{and constant}\quad p$

Consider the integrals $$ \int_{0}^{2}{k^{p}x^{k} \over 1 + x^{2k}}\,{\rm d}x\quad\mbox{with}\quad k = 1,2,3,\ldots\quad\mbox{and constant}\quad p. $$ For what values of $p$ do the integrands have an ...
1
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1answer
90 views

prove negative and positive everywhere for some point

Let $$f(x) = \begin{cases}x+ 2x^2 \sin(1/x),& x \neq 0 \\ 0,& x=0.\end{cases}$$ If I want to show $f'(0) = 1$. $\frac{f(h) - f(0)}{h} = 1 + 2h \sin(1/h) = 1$ then How to prove that ...
0
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1answer
95 views

What does $M$ mean in this context?

Was reading this question which posed the following: "Suppose $f:(a,b) \to \mathbb{R}$ satisfy $\left| \;f(x) - f(y) \; \right| \le M \;\left|\;x-y\;\right|^{\alpha} $ for some $\alpha \le 1$ and ...
3
votes
2answers
70 views

Proof that function has a maximal value

I am trying to prove that a function has a maximal value, the only problem is that very little is known about the function. The function is defined as follows: $f: \mathbb{R} \rightarrow \mathbb{R} $ ...
7
votes
5answers
887 views

How to show some function is constant?

Suppose $f:(a,b) \to \mathbb{R} $ satisfy $|f(x) - f(y) | \le M |x-y|^\alpha$ for some $\alpha >1$ and all $x,y \in (a,b) $. Prove that $f$ is constant on $(a,b)$. I'm not sure which theorem ...
3
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0answers
108 views

Volume of “deformed torus”

I'm trying to find explicit form of volume of "deformed torus": Suppose we have a curve $\gamma(t)$ in $\mathbb{R}^n$, $t\in[0,1]$. The curve closed and smooth : ...
0
votes
1answer
102 views

Convergence of Sequence of Real Numbers

Define a sequence of real numbers recursively as follows. Let $a_1 = 1$ and $a_{n+1} = 1 + \frac{1}{1+a_n}$. First, show the sequence is not monotonic. Second, show that $a_n \geq 1$ for all $n$ and ...
5
votes
1answer
99 views

Show inequality

I want to how nicely define the $f(x)$ for this type of question to prove the inequality use the mean value theorem $$e^x \ge 1+x ,\ x \in \mathbb{R}$$ How to choose $f(x)$ to show that ...
1
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1answer
77 views

$\,f \colon \Bbb R^2 \to \Bbb R $ be continuous map such that $\,f(x)=0\,$ [duplicate]

I am stuck on the following problem : Let $\,f \colon \Bbb R^2 \to \Bbb R $ be continuous map such that $\,f(x)=0\,$ for only finitely many values of $x$. Then which of the following options is ...
0
votes
3answers
34 views

Limit points and completeness

Suppose $y$ is a limit point of the metric space $X$. Show $Y=X\backslash\left\{ y\right\} $ is not complete. The following is my attempt at a proof. Say $Y$ is complete. Then $Y$ must be closed. ...
3
votes
2answers
69 views

Assumed fact about Borel sets

I am working on measurable sets and I have been coming across this, so called "fact" about unit circle. More precisely, some of proofs I am studying based on the following observation: There is a ...
0
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2answers
57 views

How many maps $\phi \colon \Bbb N \cup \{0\} \to \Bbb N \cup \{0\}$ are there …

I am stuck on the following problem when I was trying to solve an entrance exam paper: How many maps $\phi \colon \Bbb N \cup \{0\} \to \Bbb N \cup \{0\}$ are there with the property that $\, ...
2
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3answers
68 views

Are the convergent sequences dense in the bounded sequences?

Since it would be comfortable for something I am currently trying to prove if this would hold I wanted to ask here whether it is true that $c$ is dense in $l^{\infty}(\mathbb{N})$?
2
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1answer
48 views

$x^2-\log x = u $ asymptotic behaviour

Find the asymptotic behaviour as $u \to \infty$ of the solutions of $x^2-\log x = u$. Is there a standard method to solve this kind of problems? May the fact that we obviously know the derivative of ...
2
votes
2answers
88 views

Let $\,f \colon \Bbb R \to \Bbb R$ be a continuous function such that $|f(x)-f(y)|\ge \frac12 |x-y|$

I am stuck on the following problem that says: Let $\,f \colon \Bbb R \to \Bbb R$ be a continuous function such that $\,|f(x)-f(y)|\ge \frac12 |x-y|, \forall x,y \in \Bbb R$ . Then which of the ...
3
votes
2answers
311 views

Differentiability of $g:=f(\sqrt{x^2+y^2})$ for a $C^1$ function with $f'(0)=0$ NBHM $2008$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function such that $f'(0)=0$. Define for all $x,y\in \mathbb{R}$, $$g(x,y)=f(\sqrt{x^2+y^2})$$ Pick out the true statements ...
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1answer
48 views

To alternatively prove the theorem(*) by proving that $g^{(n+1)}(z_0)=0$ $\forall z_0\in \Bbb C$

Assume that $g=x+iy$ be an entire function. By a theorem(*), $\vert x(z)\vert \le N \vert z\vert ^n \ \ \forall z$ large enough and for constant $N\gt 0$ and for non-negative $n\in \Bbb Z$ ...
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1answer
44 views

Pulling Operator Inside Integral

Say $Y$ is a Banach space and you have a family of continuous/bounded operators $L_{x}: Y \rightarrow Y$ for $x\in \mathbb{R}$ and say you have an bounded, smooth map $f(x):\mathbb{R}\rightarrow Y$. ...
1
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1answer
64 views

How to argue range of a function using intermediate value theorem

Let $g(x)= 99x^3 + 999x^2 + 9999$, where $x$ is any real number. I know the image of an interval under continuous function is still an interval and this function is unbounded when $x$ goes to ...
0
votes
2answers
92 views

Cauchy integral formula in complex analysis

Assume $g$ be an entire function. And $\exists \ n\gt 0 \:and\ n\in \Bbb Z $ and also $\exists N \: and\ M \in \Bbb R$ s.t. $\forall z \in \Bbb C , \ \ \vert z\vert \ge M\ \ \: and\ \ \ \vert ...
0
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1answer
35 views

$|x-y|<|y|/2 \Rightarrow |x|>|y|/2 ?$

Although I cannot think of any counter-examples where this fails, I cannot quite understand the intuition behind the result either. If two values are quite close to each other, then this implies that ...
3
votes
1answer
57 views

Counterexample: For real functions existence of all higher order derivatives doesn't imply analycity.

In the lecture we had an example for a function $f: \mathbb R \to \mathbb R$, which is not analytic. We defined, that a function is said to be analytic at some point $x_0$ if a Taylor series expansion ...
1
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0answers
71 views

Question on Sigma Algebras

I have the measure spaces $(\mathscr{X},\mathscr{A},\mu_1), (\mathscr{Y},\mathscr{B},\mu_2), and(\mathscr{Z},\mathscr{C},\mu_3)$. I take the $\sigma$-algebra ...
1
vote
2answers
60 views

Pointwise convergence of a series

Describe the set of all points $x$ where the series $\sum_{n=0}^{\infty} 2^n/(1+x^n)$ pointwise converges. Does it uniformly converge on $[3,5]$? Where to start. Usually I fix $x$ and take it out of ...
0
votes
1answer
37 views

Finding max value

Which approach would be the best to take in order to calculate the max value of $a$, where $a\ \log(x+1)$ (blue) at no point exceeds $\sqrt{x}$ (red)?
4
votes
2answers
134 views

Is there always an injective map from a space in its dual space?

Today our teacher said that dual spaces are "big" and told us that this is a consequence by Hahn-Banach's theorem. So I was wondering whether the dual space of a space is always "bigger" or equal ...