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, integration through the fundamental theorem of calculus.

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1answer
40 views

Desperate for help showing integral is analytic

I don't have a prodigious background in analysis and would like some help... Is the integral $$f(t) = \int_{1}^{t}\sqrt{s} ds$$ analytic? Clearly, the places where $\sqrt{s}$ fails to be analytic ...
0
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1answer
58 views

Transforms carry converge sequences to converge ones

Let's consider a linear transformation $T$ of sequences: Suppose $\{c_{jk}\}$ is a matrix of complex numbers. For each sequence $\alpha=\{a_n\}$ made up of complex numbers, $\beta=T\alpha$ is a ...
4
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1answer
128 views

Horn and spindle tori

I was trying to prove that the horn torus and the spindle torus are not manifolds by definition(locally diffeomorphic to some Euclidean space.). I have no idea how to do this, but I attempted it in ...
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1answer
196 views

Cauchy Criterion for Sequences as opposed to Series

So through as I've been venturing through baby Rudin I came upon his definition of a cauchy sequence: A sequence $\{ p_n \}$ in some metric space $X$ is said to be cauchy if $$ \forall \; \epsilon ...
6
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1answer
205 views

Derivative map of the diagonal inclusion map on manifolds

I was trying to work through a problem(#10 of $\S$1.2) in Guillemin and Pollack's book $\textit{differential topology. }$ The problem is given as follows. Let $f: X\longrightarrow X\times X$ be ...
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4answers
138 views

A basic question on the derivative of a continuous function

Is the following condition necessary for the existence of derivative of a continuous function at point $x$: $$\lim_{h \to 0^+}\frac{f(x+h)-f(x)}{h} = \lim_{h \to 0^-}\frac{f(x)-f(x+h)}{h}$$
2
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1answer
229 views

$L^p$ integrable but not $L^q$ integrable

Does there exist a continuous function on $[0, \infty)$ such that it is in $L^p(0,\infty)$ for some $p\in [1,2]$ but is not in $L^q(0,\infty)$ for any $q\in (2, 2/(2-p))$? Thanks!
2
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0answers
55 views

Uniqueness for Integral Transform

What can be said about the uniqueness of the following integral transformation: $ (Tf)(u) = \int_0^{\infty} f(t)G(tu)dt$ defined for all $u\geq 0$, where the kernel $G(z) \in [0,1]$ for all ...
2
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1answer
46 views

Show that the boundary of each of the sets is contained in $b(A)\cup b(B)$ ..

Let $A$, $B$ be subsets of $\mathbb{R}^n$. Show that the boundary of each of the sets $$A\cap B, \quad A\setminus B, \quad A\cup B$$ is contained in $b(A)\cup b(B)$. Hint: ...
2
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1answer
179 views

If a series is absolutely converge then the series can be regroup with changing their order?

I am just thinking about why this is true. Can I change it to Q1. If a series is convergent then the series can be regrouped without changing the order of terms. For example the sum of $(-1)^n$ is ...
0
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1answer
50 views

Show that the boundary of each of the sets is contained in $b(A)\cup b(B)$

Let $A$, $B$ be subsets of $\mathbb{R}^n$. Show that the boundary of each of the sets $$A\cap B, \quad A\setminus B, \quad A\cup B$$ is contained in $b(A)\cup b(B)$. Hint: ...
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2answers
71 views

Differential equation with start condition and maximum capacity

The house with 1000 people have got 1 infected with an unknown virus. The number of students who are newly infected is proportional to the number of healthy product of current and the current number ...
1
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1answer
60 views

Upper Bound of Sobolev norm by $L_2$ norm

A Paper by Madych and Potter states that if a function $f\in W_2^k(\mathbb{R})$ has evenly spaced zeroes (i.e. if $Z(f):=\{x:f(x)=0\}$, is such that $\underset{y\in\mathbb{R}}\sup ...
4
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6answers
204 views

What is $\displaystyle\lim_{n\rightarrow\infty}\left(\frac{n-x}{n+x}\right)^{n^2}$?

What is $$\lim_{n\rightarrow\infty}\left(\frac{n-x}{n+x}\right)^{n^2},$$ where $x$ is a real number. Mathematica tells me the limit is $0$ when I put an exact value for $x$ in (Mathematica is ...
4
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2answers
281 views

A (not necessarily continuous) function on a compact metric space attaining its maximum.

I am studying for an exam and my study partners and I are having a dispute about my reasoning for $f$ being continuous by way of open and closed pullbacks (see below). Please help me correct my ...
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0answers
68 views

Contraction mapping proof

I am trying to show that an iterative algorithm converges to a unique point in an interval. I reduced the problem to the following: Show that $f(x) = \frac{ax^{b+1}}{c(b+1)} + \frac{2d}{ecx} = x$ ...
3
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0answers
64 views

asymptotic growth of entire functions with positive taylor coefficients

I have an entire function $g(x)$ with taylor coefficients that go to zero at a ridiculously fast rate and I am trying to bound it with another entire function $f$ with similar properties but with the ...
4
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1answer
84 views

Is there a real number, that is proven to be normal in every base, whose digits can be enumerated by an algorithm?

To clarify, I mean every natural number base $b$ where $b \geq 2$. If so, what is the algorithm to generate the number (and what is the number, if it has a name)?
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6answers
427 views

Evaluating the series $\sum\limits_{n=1}^\infty \frac1{4n^2+2n}$

How do we evaluate the following series: $$\sum_{n=1}^\infty \frac1{4n^2+2n}$$ I know that it converges by the comparison test. Wolfram Alpha gives the answer $1 - \ln(2)$, but I cannot see how to ...
3
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2answers
113 views

$f:\mathbb{R}\to\mathbb{R}$ is continuous and $\int_{0}^{\infty} f(x)dx$ exists

$f:\mathbb{R}\to\mathbb{R}$ is continuous and $\int_{0}^{\infty} f(x)dx$ exists could any one tell me which of the following statements are correct? $1. \text{if } \lim_{x\to\infty} f(x) \text{ ...
4
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2answers
129 views

Young's inequality for three variables

Let $x, y, z \geq 0$ and let $p, q, r > 1$ be such that $$ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1. $$ How can one show that under these hypotheses we have $$ xyz \leq \frac{x^p}{p} + ...
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1answer
63 views

Equality of expressions containing supremum of some double sums

I am doing a project related to operator norm and some double sequences. In the course of proving some results, I encounter the following expressions: $\displaystyle \|\alpha\| := \sup_{\|x\|_p=1} ...
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0answers
202 views

Outer measure: countable subadditivity proof

We have a space $X, \mathcal{T} \subset \mathcal{P}(X)$ (system of all subsets of $X$) and a set function $\tau: \mathcal{T} \to [0,\infty)$. For a set $A \subset X$ we define the outer measure: ...
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1answer
81 views

Given $\int _0 ^{+ \infty} \frac{1}{(1+x)^a} =1, \ \ a =?$

Given $\int_0^{+ \infty} \frac{1}{(1+x)^a} =1$ what is the value of $a$?. I know that $\int_0^{+ \infty} \frac{1}{(1+x)^2} =1$. Are there any other solutions? Could you help me?
2
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1answer
56 views

What happens outside radius of convergence

A real power series $\sum_{n=0}^\infty a_n z^n$ has radius of convergence $R$. I am able to prove that for any real number $r>R$, the sequence $|a_n|r^n$ must be unbounded. Must it also tend to ...
3
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2answers
91 views

Integral inequality, $f$ continuous, increasing function

Let $f$ be a continuous, increasing function on $[a,b]$, $c$ is the middle of $[a,b]$. Prove that $\frac{f(a)+f(c)}{2} \le \frac{1}{b-a} \cdot \int _a ^b f(x)dx \le \frac{f(b)+f(c)}{2} $ . Could ...
4
votes
6answers
451 views

Is this series convergent? $\sum_{i=1}^{\infty} \frac{(\log n)^2}{n^2}$

$\sum_{i=1}^{\infty} \frac{(\log n)^2}{n^2}$ I guess it is convergent, so I apply comparsion test for this. $\frac{log^2}{n^2} < \frac{n^2}{n^2} = 1$ So it is bounded by 1 and hence it is ...
4
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1answer
149 views

Every function is the sum of an even function and an odd function in a unique way

It is known that every function $f(x)$ defined on the interval $(-a,a)$ can be represented as the sum of an even function and an odd function. However How do you prove that this representation is ...
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5answers
645 views

$f'(x)=f(x)$ and $f(0)=0$ implies that $f(x)=0$ formal proof

How can I prove that if a function is such that $f'(x)=f(x)$ and also $f(0)=0$ then $f(x)=0$ for every $x$. I have an idea but it's too long, I want to know if there is a simple way to do it. Thanks! ...
2
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1answer
291 views

List the set of points of discontinuity of piecewise function [closed]

List the set of points of discontinuity of $f:(0,\infty)\to\mathbb R$, defined by $$f(x)=\begin{cases}x-[x]\text{ if [x] is even}\\1-x+[x]\text{ if [x] is odd}\end{cases}$$
3
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2answers
84 views

continuous function on a set of lines through a point

This came up in class once: Suppose we have a set $R \subset \mathbb{R}^2$ and $x \in \mathbb{R}^2$. If we define a function $A$ on all the lines $l$ going through $x$ such that $A(l)$ gives ...
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votes
3answers
112 views

How to test the convergence of the series $\sum_{n=1}^\infty n^{-1-1/n}$?

How to test the convergence of the series $\displaystyle\sum_{n=1}^\infty\frac{1}{n^{1+1/n}}?$ Help me. I'm clueless.
3
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1answer
104 views

Mean value theorem for essentially bounded functions

I have the problem with the following: Let $f \in L^{\infty}(\mathbb{R}_+)$ is the mean value theorem in the following form: Let $0 \leq a < b < \infty$, then $\int_{a}^{b} |f| \ d\mu ...
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2answers
104 views

A general rule for L'Hôpital's Rule

In my text there are as many as $11$ theorems dealings with the different cases of L'Hôpital's Rule and due to the quantitative heaviness I often forget rules during problem session. Isn't there a ...
26
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2answers
680 views

A beautiful limit involving primes and composites

I observed the following limit empirically. Let $p_n$ be the $n$-th prime and $c_n$ be the $n$-th composite number then, $$ \lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^{n}\frac{p_n c_n}{p_n c_n + ...
1
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1answer
182 views

Is $\frac{\sin x^2}{\sin^2 x}$ uniformly continuous on $(0,1)$? Is my proof correct?

I'm said to check if $\dfrac{\sin x^2}{\sin^2 x}$ is uniformly continuous on $(0,1)$ I can see that $\dfrac{\sin x^2}{\sin^2 x}$ is continuous on $(0,1].$ So $\displaystyle\lim_{x\to1-}\dfrac{\sin ...
3
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0answers
80 views

A question about Egoroff theorem on a family of functions

Is it possible to construct a family of measurable functions defined on a finite E that convergence is pointwise a.e. but Egoroff theorem does not hold? I was hinted that it's relate to ...
2
votes
1answer
389 views

An example of a generalized Cantor set with positive Lebesgue measure [duplicate]

I want to know if there exist a set $ X\subset \mathbb R$ such that $X$ is $i)$ Perfect $ii)$ Compact $iii)$ Has empty interior $iv)$ Totally disconnected $v)$ Is not countable But $X$ has ...
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2answers
55 views

Existence of a function satisfying given conditions

I was going through the topic of $Function$, its boundedness, continuity etc. I got a problem. Does there exist a function defined on the closed interval $[a,b]$ which is.... 1. bounded; 2. takes ...
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2answers
842 views

Showing that $f(x)=x\sin (1/x)$ is not absolutely continuous on $[0,1]$

On the interval [0,1]. Define $f(x)=x\sin(1/x)$ for $x\in(0,1]$ and $f(0)=0$. I didn't work out the exact details but I'm pretty sure that then $$\Big |\int_0^xf'(t)dt\Big |=\infty,$$ due to a ...
3
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4answers
167 views

Whether the given series are convergent or divergent:

Whether the given series is absolutely convergent, conditionally convergent or divergent: $a)\displaystyle\sum_{n=1}^{\infty}(-1)^n\frac{\cos nx}{n^2}:|(-1)^n\frac{\cos ...
0
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1answer
215 views

Convex homogeneous function

Prove (or disprove) that any CONVEX function $f$, with the property that $\forall \alpha\ge 0, f(\alpha x) \le \alpha f(x)$, is positively homogeneous; i.e. $\forall \alpha\ge 0, f(\alpha x) = \alpha ...
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1answer
296 views

Let $f(x,y,z)$ be a convex function. Is the reciprocal convex?

Suppose that $f(x,y,z)$ be a convex function. Prove $\frac{1}{f(x,y,z)}$ is convex. Or give an example of $f$ where $1/f$ is not convex. For example, I know that $f(x,y,z)=(1+x^2)\sqrt{1+y^2+z^2}$ ...
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2answers
115 views

homogeneous nonlinear functions

Give an example of degree one positively homogeneous function, (i.e. a function $f$, such that $\forall \alpha\ge0, f(\alpha x) =\alpha f(x)$) that is not linear, and $f: \mathbb{R} \to \mathbb{R}$.
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1answer
90 views

Prove that $f(x) = \sum_{n=1}^{\infty} {\frac{x^2 \sqrt{n} + 1 }{ 2n^2 + x } } $ is continuous in $(-1,2)$.

I was asked this question: Prove that $f(x) = \sum_{n=1}^{\infty} {\frac{x^2 \sqrt{n} + 1 }{ 2n^2 + x } } $ is continuous in $(-1,2)$, when $x \in \mathbb R$. On first thought, I said that if ...
1
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1answer
54 views

Criteria for a function to be plottable

Assume that $f$ is a real function. My question is how can one decide if $f$ is plottable or not? My assumption is that $f$ must be of class $C^1$, but I am not aware of such a result. My assumption ...
4
votes
4answers
416 views

What is the Idea of a Relative Open Set?

My real-analysis text gave the following defintion: Let U be a subset of E. U is open relative to E if for $\forall t \in U$, $\exists \epsilon$ such that $N_\epsilon(t) \cap E \subset U$. ...
1
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0answers
60 views

Fourier transform for even function

How find fourier transform for $$f(x) = {\mathcal{X}}_{[- \frac{1}{2}, \frac{1}{2}]} \cdot \cos^n{ \pi x}, n \in \mathbb{N} , x \in \mathbb{R} $$ where $$ {\mathcal{X}}_{[- \frac{1}{2}, ...
6
votes
7answers
269 views

$n^2\log\left(1+\frac{1}{n}\right)\to 1$ is false

How to show that $n^2\log\left(1+\dfrac{1}{n}\right)\to 1$ is false? I have to show that $\left(1+\dfrac{1}{n}\right)^{n^2}$ doesn't tend to $e.$
0
votes
2answers
285 views

Question about absolute continuous functions (preservation of null sets)

I'm trying to prove that a function $ f:[a,b] -> \mathbb{R} $ is absolutely continuous iff $ \mu(A) = 0 \implies \mu( f(A)) = 0$ for all such $A \subseteq [a,b]$. I'm quite stuck. I'm trying to ...